From 909e62339faddfcf1b99fe4415e9cce6120ff883 Mon Sep 17 00:00:00 2001 From: Dugongue <49054532+Dugongue@users.noreply.github.com> Date: Sun, 25 Oct 2026 20:18:52 +0000 Subject: [PATCH] Close full-rank coded Gaussian terminals with one rank-two query --- .../README.md | 46 ++++ .../current-provider.md | 210 ++++++++++++++++++ .../decoder.md | 130 +++++++++++ .../typed-provider.md | 164 ++++++++++++++ .../whole-k.md | 176 +++++++++++++++ 5 files changed, 726 insertions(+) create mode 100644 research/characteristic-gaussian-provider/README.md create mode 100644 research/characteristic-gaussian-provider/current-provider.md create mode 100644 research/characteristic-gaussian-provider/decoder.md create mode 100644 research/characteristic-gaussian-provider/typed-provider.md create mode 100644 research/characteristic-gaussian-provider/whole-k.md diff --git a/research/characteristic-gaussian-provider/README.md b/research/characteristic-gaussian-provider/README.md new file mode 100644 index 0000000000..e4f9d18e6b --- /dev/null +++ b/research/characteristic-gaussian-provider/README.md @@ -0,0 +1,46 @@ +# Constructive Gaussian-provider improvements + +Dugongue, with substantial OpenAI Codex assistance from four workers. Pure informal mathematics: no numerical computation, solver, Lean check or external search was used. Local DAG, linked Lean-tree sources and retained provider notes supplied the precedents. + +**Main result:** the actual whole-K/E coded negative terminal can be closed to its full independent Gaussian rank with one rank-2 query, replacing the previous two rank-1 repair queries. The construction preserves absolute phase, works at every positive lifted width, gives explicit ordinary XOR-coordinate normalizers, and retains independent inputs, owned-storage cleanup and reciprocal orientations. + +**κ status:** no higher complete multiplication κ is proved. The new full-rank repair strictly reduces its own concave tariff, but the supplied complete whole-K/E invoice still does not yield a profitable supplier. The older rank-dropping repair also remains better under the isolated concave-moment comparison. The notes state these distinctions explicitly; none of the mathematical profiles below is advertised as a new selected record. + +## Five notes and concrete results + +1. [decoder.md](decoder.md), D1–D7: characteristic-translation symmetry exposes a second commuting Gaussian direction. For the actual terminal Q₀^(-1) P D, the sequence Qχ Q₀ followed by P^(-1) returns a full Gaussian in an explicit new binary basis. The two first factors fuse into one rank-2 child. The note proves phase, inverse/transpose, finite-precision and complete-invoice statements. +2. [current-provider.md](current-provider.md), Theorems 1–9: an exact rank-selective autonomy conversion for the actual eight-mode coupled provider; same-profile inverse/dual constructions; a complete full-rank promotion of all 632 independent inputs; and a general mixed-triangular central-promotion theorem. The physical W/Z intertwiners, rather than an unjustified logical-coordinate conjugation, drive the proofs. +3. [typed-provider.md](typed-provider.md), T1–T7: exact B/C fusion; ordinary role-subspace gauges with a defect-rank query tariff; reciprocal-dual compatibility; and sector-multiplexed queries. A complete two-field example shares one Gaussian query across complementary sectors where separately factored implementations use two. A profitable selected-provider instance is not asserted. +4. [whole-k.md](whole-k.md), K1–K8: a faithful embedding of a degenerate product-cube support into a smaller even binary ambient space. At three factors the native primitive is rebuilt in dimension 18, with four actual native child ranks 8,9,8,9 instead of 8,9,15,16. Physical owner codes and modal queries are rebuilt consistently, including changed target dimensions. Its two-rank-1 complete repair is improved by the companion decoder note. +5. This README supplies the assembly map, scope and source credit. The other four notes contain 31 numbered constructive lemmas/theorems/corollaries in total, with informal proofs, plus explicit comparisons and source-specific qualifications. This count describes this release only; it is not an inherited atlas count. + +The quantities in the notes are symbolic mathematical profiles and exact algebraic consequences. They were derived by proof, without running numerical searches or executable mathematics. Publication helpers outside this directory only transport frozen document bytes. + +## Assembly: what can already be used + +The full-rank decoder applies immediately to either the original ambient-25 whole-K/E code, retaining its own rank-73 targets, or the newly reconstructed ambient-18 code, whose three fresh stages have rank-52 targets. It replaces only the terminal repair. Replacing the old local bill by the smaller one while retaining rank-73 targets is invalid. + +For the compressed case, with v = 112³ and h = 20480, the complete closed provider is + + independent targets: 2v H₅₂, + Gaussian requests: 3v(2 H₈ + 2 H₉) + 3h H₁₈ + v H₂. + +Compared with the previous full-rank repair, its target-minus-paid moment improves by v(2 − 2^t) for every 0 < t < 1. This exact improvement is insufficient to change κ, as D7 proves from the complete invoice. Ordinary code costs, effective coefficient descriptions and field-volume precision guards remain charged. No complexity theorem is inferred from a query profile alone. + +The current eight-mode provider has a different interface. Its complete autonomous promotion returns productive profile {4:28, 7:7, 8:47, 9:550} at paid profile {1:2425, 2:412, 3:510, 4:127, 5:40}. This is a proved all-input completion, not a profitable supplier asserted from nominal helper credit. Its selected-mode offers preserve the remaining coupled outputs instead of silently discarding them. + +The sector-multiplexing theorem is a complete query-sharing algorithm for its stated operators. It can be used whenever ordinary involution-sector coefficient blocks actually occur. Scalar-only boundaries with identical sector coefficients do not automatically gain from it. No such saving is subtracted from the current provider's bill in this release. + +## Exact scope and remaining work + +The release supplies positive sufficient constructions with full informal operator proofs. It does not formalize them, emit a native implementation, establish new literature priority, or certify a new best numerical κ. The principal mathematical assembly task remains a complete profitable provider or consumer combining one of these constructions with actual independent output credit. Uniform compilation must then account for all ordinary work, coefficient access, word widths, storage and error, using the exact interface that was proved. + +Only retained local sources were available for comparison. The inspected canonical snapshot is `5e0052ca0aa3036a3afa8b0438927377c8d97d53`; no claim is made to have refreshed later upstream submissions during this no-web session. This release changes no selected frontier. + +## Predecessor credit and proof-tree connections + +The current-provider lineage includes Dillon Amburgey's PR5402, Chris R/Clear1nsight's PR5407 and PR5413 audit/completion, Ali Romman's PR5377/PR5383 relation work, Cinna Davis's PR4694/PR4698 code work, and Cao Yifan's native word. The reconstructed whole-K/E lineage includes THU New Media Research Center/PR5464, Ali Romman/PR5439 and the retained native/Clifford sources. The typed-interface lineage includes DaysSky's quotient route, Cinna Davis and the controller sources, Nick Petalas, Ryanson Jonathan and Dugongue. Credit also Jacob Alan Sussman, zxv, the scalar/compiler/analytic lineage, Harvey and van der Hoeven, and the individually cited source authors. Each note gives its relevant source-relative documents; these credits do not assert that every predecessor endorsed the new arguments. + +The linked proof tree's `Mathproof/SymplecticTI.lean`, `Mathproof/LibraryBridges/Claude/TISpTransporterFibers.lean` and `Mathproof/SymplecticTITOCIBalancedCoevaluation20260901.lean` informed explicit transporter, independent-support and invariant-subspace bookkeeping. No tensor-isomorphism algorithm is invoked as a Gaussian oracle. Classical binary-form, rank-factorization, Fourier-sector, triangular-elimination and elementary Gaussian identities remain classical; the contributions here are their stated concrete provider constructions and complete endpoint accounting. + +Recommended review order: decoder D1–D3 and D6–D7; whole-K K1–K6; current-provider Theorems 1 and 8–9; typed-provider T3 and T6–T7. Every κ-related assertion should be read together with its full independent target and paid profiles. diff --git a/research/characteristic-gaussian-provider/current-provider.md b/research/characteristic-gaussian-provider/current-provider.md new file mode 100644 index 0000000000..f63807ded2 --- /dev/null +++ b/research/characteristic-gaussian-provider/current-provider.md @@ -0,0 +1,210 @@ +# A rank-selective autonomy conversion for the actual eight-mode provider + +Dugongue, with substantial OpenAI Codex assistance. Informal mathematical proofs only; no numerical experiment, solver, executable theorem check, Lean run or external search accompanies this note. + +The positive construction is a family of actual partially decoded offers. Selecting k of the eight current mode directions returns those k original helpers as autonomous full rank-nine Gaussian jobs at the additional price of k central rank-one jobs. All 632 original inputs remain arbitrary, and the other coupled directions are written explicitly. The construction does not declare its changed main interface to be a completed profitable Gaussian supplier. No higher complete κ is established here. + +## Exact local sources and predecessor credit + +The source inspected was the stored canonical tree at commit `5e0052ca0aa3036a3afa8b0438927377c8d97d53`, without fetching or executing it. The relevant documents are: + +- `research/coupled-provider-independent-audit/capsule/repository/research/native-rank8-helper-provider/PROOF.md`, the actual PR5402 construction by Dillon Amburgey, and the surrounding PR5407 audit by Chris R/Clear1nsight; +- `research/coupled-helper-neutral-completion/README.md`, the PR5413 independent completion and its source review; +- `research/native-center-free-code/CODED-PROOF.md`, retained within that audit's capsule, and the source center-free work of Cinna Davis/PR4694 and PR4698; +- `research/native-coupled-inverse/README.md`, which supplies the actual all-input inverse with the same Gaussian profile; +- `research/eight-owned-relations/README.md`, preserving Ali Romman's PR5377/PR5383 relation mechanism and its source/pricing lineage. + +Credit Cao Yifan's native word, Ali Romman's cached relations, Cinna Davis's helper/controller work, Dillon Amburgey's coupled producer, Chris R's audit and completion, Dugongue's native and code interfaces, zxv, Jacob Alan Sussman, Ryanson Jonathan and all retained Clifford, scalar, compiler and analytic contributors. Classical rank projection, triangular inversion and linear transposition are not claimed as new mathematics. Source inspection does not supply a fresh proof-kernel certificate or an exhaustive novelty audit. + +## 1. The exact current offer + +Let V be the complex address-vector space at any positive lifted digit width. There are 112 fields in each main bank x,y and eight mode fields h. The other 400 helper fields have autonomous Gaussian endpoints and are unchanged throughout the construction below. Let D be the Gaussian on the eight noncentral address axes, Q the central-axis Gaussian, and R=DQ=QD the full rank-nine Gaussian. These are applied diagonally to their respective banks. + +The actual coupled provider M has the endpoint + + X = D x − R y + R W h, + Y = D y, + H = D(h − Z y). (1) + +It has 632 arbitrary original inputs, nine arbitrary owned inputs cleared before use and zero on return, and paid Gaussian profile + + P = {1:2193, 2:412, 3:510, 4:127, 5:40}. + +Its autonomous helper profile is + + H_aut = {4:28, 7:7, 8:47, 9:318}. + +The eight fields H in (1) are coupled and do not add eight independent Gaussian targets. + +The source supplies ordinary finite maps + + W = diag(K_p^(-1)) M_0 diag(K_m)/2, + Z = diag(K_m^(-1)) M_0^T diag(K_p)/2, + +where M_0 is a real scalar 112 by 8 matrix with M_0^T M_0=56I, and all K_p,K_m are actual real orthogonal monomial address maps. Its exact identities are + + ZW = 14I, WZ = I−C, + R W = W R, Z R = R Z. (2) + +The R on each side acts on the appropriate main or mode bank. All scalar coefficients are applied once, outside the address tensor power. Neither W nor Z is assumed to intertwine D or Q separately. That distinction is essential. + +The ordinary maps in this note mean their explicit scalar/monomial implementations. They have real nonrecursive costs. They are not Gaussian oracle calls or arbitrary dense changes of address basis. + +## 2. A selected mode projection + +Fix a coordinate subset J of {1,...,8}, let E select those coordinates, let k=|J|, and put F=I−E. Thus E²=E, F²=F, EF=FE=0. These are scalar mode-bank maps and commute with D,Q,R on the mode fields. + +Define three explicit operations on the original input/output banks: + + P_E: h ← h + E Z y, + Q_E: apply Q on H_j for each j in J, + O_E: X ← X − W E H. + +Each P_E and O_E is an ordinary invertible shear: its inverse changes the addition sign. Q_E consists of k charged rank-one jobs. No original input is initialized, identified with another input, or thrown away. + +## Theorem 1. An all-input partially autonomous offer + +Execute P_E, the unchanged coupled provider M, Q_E, then O_E. The complete endpoint is + + Y = D y, + E H = R E h, + F H = D(F h − F Z y), + X = D x − R C_E y + R W F h, (3) + +where + + C_E = I − W E Z. (4) + +All 632 original inputs are arbitrary. Exactly the selected k mode helpers are autonomous full rank-nine jobs. + +**Proof.** After P_E the mode input is h+EZy. Substitute this into (1). The mode output before Q_E is + + D(h+EZy−Zy)=D(h−FZy). + +Applying E gives DEh; applying the k central Q jobs therefore returns REh. The complementary mode output is D(Fh−FZy) and is unchanged by Q_E. + +The main output before O_E is + + Dx−Ry+RW(h+EZy) + = Dx−R(I−WEZ)y+RWh. + +The selected mode output entering O_E is REh. The intertwiner RW=WR in (2) gives W(REh)=RWEh, so subtracting WEH removes exactly that part. The remaining main term is RWFh. This proves every identity in (3), including its operator order. The selected output REh depends only on its original selected input, not on x,y or any unselected helper. ∎ + +The construction uses no interchange of D with W or Z. In particular, substituting an assumed D-intertwiner here would change the source contract. + +## Theorem 2. Exact paid and autonomous inventories + +The construction of Theorem 1 has paid profile + + P_E = {1:2193+k, 2:412, 3:510, 4:127, 5:40}, + +and autonomous helper profile + + H_E = {4:28, 7:7, 8:47, 9:318+k}. (5) + +It uses the source's nine owned fields, with additional ordinary temporary fields if the chosen explicit W/Z implementation needs them; all such extra fields can be zero on return. The source's other 400 helpers retain their original all-input endpoints and are never used as zero scratch. + +**Proof.** The only newly inserted nonordinary operations are exactly k Q calls. P_E reads y, adds its prescribed E Z image to the selected original h fields, and changes no helper outside that selected bank. It is an invertible input shear, not preparation of a supposedly dependent original input. O_E is another invertible shear. Both can be evaluated by computing their finite sums from immutable source banks into owned temporary arrays, adding them to their targets, and clearing those temporaries after their final reads. Source ownership and endpoint guarantees apply on all such inputs. Theorem 1 supplies k new autonomous R endpoints and leaves the other 400 autonomous source endpoints untouched. This proves (5). ∎ + +The positive helper-only moment change is k(9^t−1) at exponent t. This change is real as a statement about the autonomous helper inventory, but its changed main and remaining mode operators must also be consumed. It is not a complete target-minus-paid supplier moment by itself. + +## Theorem 3. The selected code is invertible, bounded and full-R compatible + +Let A_E=WEZ. Then + + A_E²=14 A_E, + C_E^(-1)=I−A_E/13. (6) + +For coordinate E, C_E is self-adjoint, equals −13 on the image of WE, and equals 1 on the kernel of EZ. It commutes with full R. Its spectral norm is at most 13 and its inverse norm is at most 1. + +**Proof.** Using ZW=14I and E²=E gives A_E²=WE(ZW)EZ=14WEZ. Multiplying I−A_E by I−A_E/13 gives I+A_E(−1−1/13+14/13)=I, proving the inverse formula on both sides. The actual real orthogonal monomial normalizers imply Z=W*, because the scalar matrix M_0 is real. Hence A_E=W E W* is self-adjoint. By ZW=14I, the map W/√14 is an isometry from the mode space into the main-bank space. Thus A_E/14 is the orthogonal projection onto the image of WE. The claimed eigenspaces and norm bounds follow. Finally, (2) and ER=RE give RWEZ=WEZR, so C_E commutes with R. ∎ + +The coefficient 1/13 in (6) is an explicit rational mathematical formula. It is not treated as a free dyadic primitive. The routines P_E,O_E themselves use the source's dyadic scalar/monomial coefficients and require no evaluation of C_E^(-1). + +## Theorem 4. The complete mixed endpoint is invertible + +The endpoint (3), including all eight mode outputs, is an isomorphism of all original inputs. A literal inverse is obtained in the following order: + + y = D^(-1) Y, + E h = R^(-1) E H, + F h = D^(-1) F H + F Z y, + x = D^(-1)[X + R C_E y − R W F h]. (7) + +The other 400 helper jobs are inverted by their own original inverse Gaussian maps. + +**Proof.** The first three equations invert the corresponding triangular equations in (3). Their right sides are known by that point, and the invertible factors D,R exist. The last equation then inverts the main X equation. Direct substitution reconstructs every original field. This also follows from T_E=O_E Q_E M P_E as a product of invertible maps on the persistent inputs. ∎ + +Formula (7) is a correctness identity; it is not the most economical way to implement the inverse. The same-price construction below uses the already supplied inverse of M rather than separately charging every occurrence in (7). + +## Theorem 5. A same-profile inverse of the full mixed offer + +Assume the retained source-bound exact inverse of M, supplied in `research/native-coupled-inverse`, with the same paid profile P and its owned-zero return. Then the inverse of T_E from Theorem 1 has exactly the same Gaussian profile P_E as its forward construction. Every operation, phase and owned-state action is specified by reversing the composition and inverting its explicit factors. + +**Proof.** Execute O_E^(-1), then Q_E^(-1), then the supplied M^(-1), then P_E^(-1). The ordinary inverse shears change only their scalar signs and preserve source fields during their last uses. Q_E^(-1) has k inverse rank-one Gaussian jobs; each is normalized by the actual even-power monomial if a positive child is required. M^(-1) has profile P by its source theorem. Composition proves T_E^(-1)=P_E^(-1)M^(-1)Q_E^(-1)O_E^(-1), preserving the same independent original dimensions. All source phases are retained in those inverse factors. ∎ + +## Theorem 6. Linear transposition supplies the full reciprocal dual + +A fixed finite linear algorithm on role fields, built from scalar ADD/SCALE, monomial address maps and Gaussian children, has an exact transpose algorithm. It uses one transposed Gaussian child for each original child, plus finite ordinary work and owned storage. In the actual Gaussian alphabet, transposing a rank-r child preserves its rank and gives an explicitly monomial-normalized rank-r child. Applying this to the inverse construction in Theorem 5 gives a full T_E^(-T) offer with profile P_E. + +**Proof.** Introduce one adjoint buffer for each version of each forward wire, all initially zero. Seed the specified output buffers with the arbitrary dual input fields. Process operations in reverse order. For z=a u+b v, add a z* to u* and b z* to v*. A monomial map U passes its adjoint by U^T, another monomial map including the exact transposed phase. A Gaussian child A passes its adjoint by A^T once, after all its downstream adjoint contributions have been accumulated. Thus shared forward wires do not duplicate a Gaussian call. COPY is handled by accumulation into its source adjoint, and an unused cleared wire has zero adjoint injection. + +The invariant is the formal bilinear pairing between wire values and wire adjoints. Every reverse replacement preserves that pairing and removes one forward instruction. At the original input wires it proves the exact transpose action. No original runtime data are needed in this reverse construction because all coefficients are fixed and the algorithm is linear. Owned buffers are explicitly cleared after their final reads; arbitrary initial owned dirt is first cleared. + +Every transpose in this construction is taken on the actual physical linear map. A bookkeeping zero retag is physical identity and remains identity; it is not replaced by an unproved nonzero frame conversion. For a physical Gaussian child, use its actual factorization U=M_L C_r M_R, with ordinary monomials and the canonical symmetric tensor Gaussian C_r. Then U^T=M_R^T C_r M_L^T has the same one rank-r request and explicit ordinary wrappers. Exact phase is retained because the operator factors themselves are transposed. The finite source inverse algorithm in Theorem 5 has precisely the original Gaussian ranks. Its transpose therefore gives T_E^(-T) with those same ranks and the same k additional central jobs. ∎ + +This is a mathematical finite-template construction with a larger but finite ordinary/owned allowance. It is not a claim that the old in-place transcript is simply reversed without treating COPY and CLEAR correctly. The dual's complete mixed endpoint is the actual inverse transpose, not a transpose relabeled by commuting an even D² through C_E. + +## Theorem 7. All positive digit widths and finite guards + +Theorems 1–6 hold at every positive lifted digit width under the source's exact physical-map hypotheses. The added ordinary preparation, decoding and transposition work has finite-template prefix and suffix norm bounds independent of that digit width, with the usual explicit coordinate-to-field rounding conversion. All additional nonrecursive work and owned storage have fixed finite bounds before the external input length grows. + +**Proof.** The source identities (1)–(2) hold under the exact native lifted operator assignment. E is a scalar role projection; it is not tensor-powered on address columns. P_E and O_E use the fixed source monomial/owner words, so they lift with exactly the original scalar coefficients. Q_E is an actual rank-one Gaussian child at that lifted width. Every algebraic composition above therefore remains valid. + +In the maximum-over-fields Euclidean norm, each monomial/child is unitary. Each scalar ADD by c has norm at most 1+|c|, and every fixed scalar program has the product of its finitely many such factors as a prefix/suffix bound. The source provider already supplies a finite such envelope. Multiplying by the extra finite P_E/O_E envelopes and the finite transpose-program envelope gives another constant depending on the template, not on digit width. Local rounding errors propagate by the sum of their norms times the appropriate suffix bound. The count of coordinate roundings and the dimension conversion must be included in any native implementation's actual precision budget; no dimension-free coordinate claim is inferred merely from a field norm. ∎ + +## 3. Honest endpoint cases and the completion question + +For k=8, F=0, so all 408 helpers are autonomous, the main block is + + X=Dx−RC y, Y=Dy. + +This is the already documented center-free coded offer. Its profile P+8H1 equals `{1:2201,2:412,3:510,4:127,5:40}`. Recovering that endpoint by a shorter block proof is not a new profile improvement. + +For k=0 all eight original modes remain coupled. If one instead redesignates all eight as owned zero inputs, (1) gives the ordinary uncoded main block and 400 independent helpers, with the unselected outputs subsequently cleared only because those fields are owned. This is the already documented relation-restricted offer. Intermediate k can similarly produce an independent-input smaller offer with a rank-k code, but its reciprocal dual must be rebuilt after that restriction: principal restriction and inverse transpose do not commute in general. + +The full 632-input family in Theorems 1–7 avoids that restriction and has an actual same-profile inverse and reciprocal dual. Its selected helpers are genuine autonomous outputs; its remaining modes are honestly coupled. A larger complete κ would require a caller that consumes this exact mixed interface, or a cheaper standardization of the rank-k code together with all remaining mode effects. No such stronger complete caller is asserted here. + + +## Theorem 8. A fully autonomous full-rank completion + +The physical rank-selective construction at k = 8 has an exact completion that returns all 224 original main inputs as separate full-rank R jobs, together with all 408 original autonomous helper jobs. Its paid profile is + + {1:2425, 2:412, 3:510, 4:127, 5:40}. + +This theorem refers to the physically standardized maps of Sections 1-2, where W and Z are actual ordinary monomial/scalar programs and C = I-WZ intertwines the full R. It is not inferred from a logical-coordinate code alone. + +**Proof.** Use the raw physical endpoint directly. First apply Q to all 112 Y fields, obtaining Y_1 = Ry. Next apply Q to the eight coupled H fields, obtaining H_1 = Rh-RZy; add ZY_1 to obtain H_2 = Rh, using the actual intertwiner RZ = ZR. Now replace X by X+Y_1-WH_2 = Dx, using RW = WR, and apply Q to all 112 X fields to obtain X_2 = Rx. This adds 112+8+112 = 232 central calls to the raw 2193 rank-one calls. The 400 already autonomous helpers are untouched. The eight formerly coupled modes become genuine independent rank-nine jobs, so the final autonomous helper profile is {4:28,7:7,8:47,9:326}. This direct completion needs only the ordinary W and Z programs: no helper pre-compensation, no coded consumer, no C inverse and no division by 13. Ordinary decoding can use fixed temporary buffers which are cleared before use and after their last read. Thus arbitrary dirty owned inputs do not contaminate the outputs, and all owned buffers return to zero. + +The full linear action is diag(R,R) on the two main banks and the inherited autonomous helper action on all 408 helpers. Consequently all 632 original directions are independent and preserved. Applying the inverse construction and the physical transpose construction of Theorems 5-6 gives both reciprocal orientations with the same child ranks. Fixed ordinary code, monomial normalization, storage and precision bounds are retained as in Theorem 7. No inverse of C and no division by 13 is needed. No commutation between C and D or Q is used. + +The complete productive profile is {4:28,7:7,8:47,9:550}. This complete profile must be compared directly with its paid profile in the actual supplier inequality. The earlier coded endpoint does not independently justify crediting 224 autonomous rank-eight main jobs. Both profiles have rank mass 5487. For the relevant concave pricing range 0 < t < 1, cancel 28 rank-four jobs on each side. Every remaining paid rank is at most 5 and every remaining target rank is at least 7. Their equal remaining rank mass, together with r^t = r r^(t−1), proves that the paid moment is strictly larger than the productive moment. Thus this complete promotion is a valid interface construction, but does not improve κ. No new complete multiplication exponent or absence of overlap with prior submissions is asserted. + +## Source-word audit of the one-sided extraction + +The same completed coded endpoint, restricted to an owned x = 0 bank, gives X = -RCy and Y = Dy. The ordinary post-map -C^-1 on X therefore gives Ry. This is 112 independent full-rank jobs, not 224: X and Y both depend on the same 112 arbitrary y inputs. The 408 helper jobs remain independent. The mathematical inverse code involves 1/13; it must be implemented with an explicitly admitted finite scalar approximation or another paid representation if the primitive alphabet permits only dyadic constants. No division oracle is assumed. + +Merely pruning transports before the first nonzero X write and after the last use of a discarded Y output does not supply a cheaper Gaussian profile in the inspected backward native word. Its first X83 interaction is a rank-zero warp followed by an ADD from Y126, at source events 6275 and 6277. Its last Y rank-one calls are still read by productive helper computations: for example Y159 is transformed at event 15242 and read again through event 15270; the last main Y read occurs at event 15272. There is no later paid Y transport to delete by the terminal-discard rule. These literal chronology observations refute the proposed automatic boundary saving; they are not an exhaustive impossibility result for internal rewrites. The positive full-rank completion above avoids relying on such a saving. + + +## Theorem 9. Central promotion of mixed triangular interfaces + +Let a finite ordered bank have arbitrary independent inputs u_1,...,u_m and exact outputs + + v_i = D_i u_i + sum_{j 1 and a positive digit width f. An address is an f-tuple of vectors in F₂^N. For a binary direction v, let X_v translate one address column by v. Write + + a = (1+i)/2, b = (1-i)/2, C_v = a I + b X_v. + +The superscript [f] means applying the displayed one-column operator separately in all f columns, that is its tensor lift. Thus C_v^[f] is not a I + b times the simultaneous translation on every column. Put + + Q₀ = C_e₀^[f], + D = (C_e₁ ... C_e_(N−1))^[f], + R = Q₀D, + χ = e₀ + ... + e_(N−1), Qχ = C_χ^[f]. + +A rank-r Gaussian child H_r acts on r independent binary directions in every address column, with identity on all other address coordinates. An explicit invertible binary coordinate change is an ordinary address permutation; conjugating a canonical child by it does not add Gaussian requests. Its real finite nonrecursive work is retained. + +Let P be an explicitly ordinary invertible finite code on a bank of owner fields. Assume it commutes, separately for each column j, with translation X_(χ,j) on every field. The two actual terminal operators to be decoded are + + W₋ = Q₀^(-1) P D, W₊ = P D Q₀. (1) + +Equation (1) is a hypothesis about the literal physical code, not a statement obtained by identifying its logical coordinates. The source application below establishes it from retained whole-K/E interfaces. This note never assumes that P commutes with Q₀ or D. + +## D1. Exact translation square and inherited column symmetry + +**Lemma.** C_v² = X_v, with no scalar phase. If a one-column real monomial L commutes exactly with the full local Gaussian R_s on a set of s axes, then L commutes with translation by that local set's characteristic vector. When L is lifted columnwise and fixes all other axes, it commutes with X_(χ,j) for every global column j. Finite scalar mixing between owner fields preserves this property. + +**Proof.** Since X_v² = I, expand C_v². The identity coefficient is a²+b² = 0 and the translation coefficient is 2ab = 1. The coordinate Gaussians commute, so R_s² is the product of the translations on the local axes, namely translation by their sum. An L commuting exactly with R_s commutes with its square. It fixes each axis outside its local set, so it also commutes with translation by the sum of the remaining global axes. Multiplication gives the desired global translation. A columnwise lift gives this argument independently in every column. Scalar owner coefficients commute with a common address translation, and finite products, sums and inverses of commuting operators retain that commutation. ∎ + +Commuting only with R^[f]² would give a weaker simultaneous-column statement and would not justify the next lemma. The columnwise premise above is essential. + +## D2. A second Gaussian direction in the code's commutant + +**Lemma.** Under the starting contract, P Qχ = Qχ P. + +**Proof.** In column j, C_(χ,j) = a I + b X_(χ,j). The hypothesized commutation with the translation therefore gives commutation with this one-column Gaussian. Multiply those identities over all f columns. ∎ + +This is a constructive commutant element. It does not move the source's distinguished central line to the characteristic vector, which would be an invalid shortcut for the orthogonal normalizers in question. + +## D3. The full-rank negative decoder + +**Theorem.** Apply, in chronological order, Qχ Q₀ and then the ordinary reversed inverse code P^(-1) to the negative terminal W₋. The result is exactly Qχ D. It is an ordinary monomial conjugate of the full rank-N Gaussian R, with no phase ambiguity. The extra paid Gaussian tariff is one H₂ per negative owner field. The positive terminal requires only the ordinary decoder P^(-1) and returns R. + +**Proof.** Operator products act right to left. Lemma D2 gives + + P^(-1)(Qχ Q₀)(Q₀^(-1) P D) + = P^(-1) Qχ P D + = Qχ D. (2) + +Every equality retains its scalar phase. To implement the first factor in (2) as a single H₂, use the binary matrix A with ordered columns + + e₀, χ, e₂, ..., e_(N−1). + +It is invertible: the e₁ coordinate forces the coefficient of χ to vanish in any dependence, and the remaining coordinate columns are independent. For the ordinary permutation Π_A of address labels, direct action on a basis vector gives + + Π_A X_v Π_A^(-1) = X_(Av), + Π_A C_v Π_A^(-1) = C_(Av). + +Hence Qχ Q₀ = Π_A (C_e₀ C_e₁)^[f] Π_A^(-1), a single canonical rank-2 child between explicit ordinary permutations. + +Likewise the binary matrix B with columns χ,e₁,...,e_(N−1) is invertible, since the e₀ coordinate first forces the coefficient of χ to vanish. The same conjugation identity gives + + Qχ D = Π_B R Π_B^(-1). (3) + +After the decoder in (2), apply Π_B^(-1) on outputs and Π_B on the corresponding independent original input before executing (1). The complete operator is then exactly R, not merely an output with the same Gaussian rank. Applying the input permutation changes neither the independence nor the arbitrary values allowed for original inputs. The positive terminal satisfies P^(-1)W₊ = DQ₀ = R. The two banks are independent, so the input/output normalizers on the negative bank do not change the positive bank. ∎ + +Both coordinate permutations have elementary descriptions. A maps y₁=x₁, y₀=x₀+x₁ and y_j=x_j+x₁ for j≥2; B maps y₀=x₀ and y_j=x_j+x₀ for j>0. They operate in each digit column. This gives literal finite XOR-based permutations, rather than an unspecified linear-algebra oracle. + +## D4. Exact reciprocal orientations and owned storage + +**Theorem.** The decoded complete construction admits an inverse and a reciprocal transpose construction with the same Gaussian ranks. All original fields remain independent. Ordinary temporary storage may start dirty, is initialized before use and can be zero on return. Unrelated fields are preserved. + +**Proof.** Reverse the complete word of abstract invertible main factors and use their supplied inverse routines. In particular, use the explicit ordinary P/P^(-1) code; do not try to reverse scratch initialization, COPY or CLEAR instructions, which need not be invertible. The adjoint construction below instead transposes the versioned linear straight-line DAG. The inverse of a Gaussian C_v is X_v C_v, since C_v² = X_v and C_v⁴ = I; thus its inverse has the same paid rank with an explicit ordinary translation. Coordinate normalizers invert as address permutations. This gives the literal inverse, including its ordinary coefficients. + +For a transpose, reverse the linear straight-line computation into separate adjoint field buffers. Scalar accumulation transposes by the same coefficient, a monomial address map by its transposed permutation and phase, and each symmetric canonical Gaussian by itself. The conjugated-child boundary permutations are transposed in reversed order. Seed output adjoints with arbitrary dual inputs, accumulate the input adjoints, and clear working buffers after their final reads. Applying this to the inverse gives the reciprocal transpose. Each child is used once with the same rank. + +For a forward implementation, compute any ordinary scalar combinations from source snapshots into initialized owned buffers, accumulate the destination and clear the no-longer-needed snapshots. Initialization of owned scratch is allowed by the starting source contract; no independent productive input is initialized or erased. The finite inverse and transpose constructions retain this distinction. ∎ + +This theorem uses the usual finite-field-array linear algorithm model. It does not grant the unpriced inverse of an unspecified dense address operator: P and both permutation normalizers have already been supplied as ordinary words. + +## D5. Finite precision at every positive width + +**Theorem.** For a fixed finite provider template satisfying the contract, the decoder has a finite all-width error envelope. If its ordinary coefficients have effective O(p)-bit descriptions, an O(p)-bit guard budget suffices when the logarithm of the field volume, the logarithm of the admitted input-magnitude envelope, and requested output precision are O(p). The square root of the field volume must be charged when a coordinate-rounding guarantee is derived from a Euclidean field-norm bound. + +**Proof.** Let there be M rounded primitive sites and at most V scalar coordinates at any such site. Let K bound every relevant suffix operator norm, including the actual finite ordinary inverse P^(-1). The exact monomial permutations and Gaussian children have Euclidean norm one; finite ordinary coefficients supply a finite K independent of lift width when they are fixed scalar combinations of finitely many orthogonal monomials. If coordinate errors at a site are at most 2^(-B), the field Euclidean error is at most √V 2^(-B). By the telescoping perturbation identity and the triangle inequality, the final error is at most + + K M √V 2^(-B). + +Thus B ≥ P_target + log₂(KM√V) + 2 is sufficient for output error below 2^(-P_target). This retains the field-volume term and includes all sites at which an actual implementation rounds. If the whole provider has O(p) sites or a polynomial-size width-dependent schedule, its log M is also O(p). The stated hypotheses therefore give B = O(p). Coefficient approximation errors are included as extra sites with their actual input magnitude bounds. The stated input-magnitude hypothesis and the fixed finite coefficient/norm envelope add O(p) guard bits for their scaling; absent that hypothesis, the extra input-range bits must be charged separately. No exact division is free. ∎ + +This is an informal sufficient precision theorem under explicit parameter-description hypotheses, not an emitted native implementation or an independent compiler acceptance certificate. + +## D6. Application to the retained whole-K/E word + +**Theorem.** The ordinary physical codes E_i of the retained whole-K/E construction satisfy the column symmetry used in D1. For its three fresh-stage endpoints (1), the two full-rank negative repairs previously billed as 2 H₁ per owner may be replaced by H₂, with unchanged full independent target ranks. In the reconstructed ambient-18 provider of `whole-k.md`, the complete profile becomes + + targets: 2v H₅₂, + paid: 3v(2 H₈ + 2 H₉) + 3h H₁₈ + v H₂, + v = 112³, h = 20480. (4) + +**Proof.** In each stage, the physical code is an ordinary scalar owner matrix conjugated by the diagonal bank of actual real monomial normalizers L_i. Those normalizers commute exactly with the complete Gaussian on that stage's central-plus-fresh axes, separately at one-column level; the source's anchor identities retain the phase. Lemma D1 therefore applies to each L_i, to its inverse, and to the scalar owner recodings, including their rational E corrections. A stage fixes all fresh axes belonging to other stages. Consequently each stage code commutes with the global characteristic translation in every column. The chronological product P = E₃ E₂^(-1) E₁ and its ordinary inverse have the same property. + +Here is also the literal three-stage derivation. Put H₀ = diag(I,Q₀), let L = [[1,0],[1,1]] and U = [[1,−1],[0,1]], and write A_s = E_s^(±1) D_s for the appropriate forward or backward stage. The corrected source words are H₀^(-1) diag(A_s,A_s) L H₀ and H₀^(-1) diag(A_s,A_s) U H₀. The three-stage product is H₀^(-1) diag(A₃A₂A₁,A₃A₂A₁) J H₀, with J = LUL = [[0,−1],[1,0]]. The final ordinary J^(-1) makes its two blocks Q₀^(-1) A and A Q₀. Freshness means D_s commutes with every other-stage E_t, hence A = E₃ E₂^(-1) E₁ D₁₂₃ = P D₁₂₃. This proves (1) in the actual coordinates, with genuinely ordinary P; no hidden Q₀^(-1) E Q₀ recoder remains inside P. D3 closes these terminals. In the ambient-18 reconstruction each stage contributes 17 fresh noncentral axes. Three stages plus the common central axis give N = 52. There are v independent negative fields, so the new decoder costs v H₂. The three local calls retain exactly the K6 paid profile in `whole-k.md`; this proves (4). The source field ownership and D4 preserve all independent targets. ∎ + +The same replacement applies to the original ambient-25 construction at its own unchanged full output rank 73. It must not be combined with the compressed local bill while retaining rank 73 targets. + +## D7. The exact improvement and its scope + +**Corollary.** For every concave pricing exponent 0 < t < 1, replacing the two rank-one repairs by the full-rank decoder of D3 improves target-minus-paid power moment by + + v (2 − 2^t) > 0. (5) + +It does not improve the complete κ supplied by the explicit invoice (4). + +**Proof.** The independent target profile is unchanged, and the paid repair changes from 2v to v 2^t, proving (5). The target rank mass in (4) is 104v. Its paid mass is 102v + 54h + 2v = 104v + 54h. Every paid rank is at most 18 and is strictly below 52. For 0 < t ≤ 1, each paid rank r obeys r^t ≥ 52^(t−1)r. Summing shows its paid moment exceeds the target moment 2v 52^t, since h > 0. Thus this supplied complete construction is not a profitable supplier for that exponent range. ∎ + +There is a further important comparison. The older single-H₁ repair can retain only rank N−1, whereas D3 retains rank N. For N > 2 and 0 < t < 1, + + [N^t − 2^t] − [(N−1)^t − 1] + = N^t − (N−1)^t − (2^t − 1) < 0, + +by strict decrease of the increments of x^t. Thus the new full-rank interface is cheaper than the previous full-rank repair, but the rank-dropping option still has the larger concave moment considered in isolation. This note does not count the full-rank closure as an already funded κ improvement. + +## Predecessor credit and local proof-tree use + +The starting terminal derivation and ordinary E codes come from `research/odd-e8-whole-k-native/PROOF.md`, `research/odd-e8-lowrank-e-native/E-PROOF.md`, and the retained `research/central-q-decoder/PROOF.md` and `E-COMPOSITION.md`. Credit THU New Media Research Center/PR5464, Ali Romman/PR5439, Dugongue, Dillon Amburgey, Cinna Davis, Ryanson Jonathan, Cao Yifan, Jacob Alan Sussman, Chris R/Clear1nsight, zxv and all retained native, Clifford, compiler and analytic contributors. These source relationships were read locally; this is not a claim of exhaustive literature priority. + +The linked symplectic/TI sources `Mathproof/SymplecticTI.lean` and `Mathproof/LibraryBridges/Claude/TISpTransporterFibers.lean` informed the use of an explicit intertwiner and actual operator stabilizer, rather than only a scalar invariant or a rank match. Their existence does not formalize this decoder. The binary translation and elementary Gaussian identities are classical; the contribution here is this explicit full-rank terminal application and its honest complete invoice. diff --git a/research/characteristic-gaussian-provider/typed-provider.md b/research/characteristic-gaussian-provider/typed-provider.md new file mode 100644 index 0000000000..cc190c322d --- /dev/null +++ b/research/characteristic-gaussian-provider/typed-provider.md @@ -0,0 +1,164 @@ +# Typed Gaussian providers, subspace gauges and sector-multiplexed queries + +Dugongue, with substantial OpenAI Codex assistance. Informal mathematics based only on local source and proof-tree scans. No numerical experiment, solver, web search or Lean check was used. The constructions below have complete all-input operator proofs on their stated domains; they do not assert an unproved profitable provider or a larger numerical κ. + +## Sources and actual starting contract + +The actual local sources are `research/native-dual-suppliers/HETEROGENEOUS-CONTROLLER.md`, `FREE-ENDPOINT-GAUGES.md`, `DUAL-PROVIDER.md` and `ROUND3-IMPROVEMENT.md`; the scalar event definition in `research/kappa-rebuild/bit/complex_controller_factor.py` was read as text and was not executed. The preceding mathematical continuation supplies `boundary.md` and `consumer.md`. The linked symplectic balanced-coevaluation and mode-support notes motivate reciprocal-dual and invariant-subspace bookkeeping; no fresh formal verification is inferred from those sources. + +The admitted B and C local words have 112 independent main pairs, 408 independent helper fields, and respectively 56 A plus 56 B or C ports. Their exact main types are + + A = D [[1,0],[Q^(-1),1]], + B = D [[1,0],[1,Q]], + C = D [[Q,0],[1,1]], + +where D acts on seven fresh axes and Q on the common central axis. Helpers and paid Gaussian jobs retain their individually stated profiles. Arbitrary main-pair permutations place the exceptional ports but do not identify helper inputs. Backward calls use the actual inverse-transpose endpoints with their even-power monomial corrections. + +Credit Cao Yifan for the retained native round-three word, DaysSky for the quotient construction, Cinna Davis and the controller lineage, Nick Petalas and Ryanson Jonathan for endpoint analysis, Dugongue for transported/repaired interfaces, zxv and Jacob Alan Sussman for compiler foundations, and all retained Gaussian/Clifford/source08 contributors. Ordinary rank factorization, projector decomposition, nullity and elementary matrix identities are classical. The new applications here are the precisely stated fusion, projection-gauge and sector-query interfaces. + +## T1. Complete exceptional-port extraction and two-call fusion + +Let S_y(t)=[[1,0],[t,1]], and suppress the common commuting D factor for a moment. + +**Theorem.** Exceptional B and C ports have the exact ordinary-boundary extractions + + S_y(−1) B = D diag(1,Q), + C S_y(−1) = D diag(Q,1). + +For two calls on the same pair, with arbitrary commuting fresh factors D_1,D_2, + + C_2 B_1 = D_2 D_1 Q S_y(2Q^(-1)), + S_y(−1) B_2 C_1 = D_2 D_1 Q S_y(Q^(-1)). + +**Proof.** In the first identity subtract the first output row of B from its second. In the second replace the first input contribution in C's second row by its difference with the second input. Both are the stated elementary shear and give the displayed diagonals on arbitrary independent inputs. + +For the two-call products multiply the two explicit matrices: + + [[Q,0],[1,1]] [[1,0],[1,Q]] = [[Q,0],[2,Q]], + [[1,0],[1,Q]] [[Q,0],[1,1]] = [[Q,0],[1+Q,Q]]. + +Factor out the common Q; in the second product the remaining shear is S_y(1+Q^(-1)), whose ordinary S_y(1) factor is removed on the left. Restore the commuting D factors. ∎ + +The direct B extraction is inherited predecessor work, including the PR4660 extraction discussion; it is not claimed anew. The combined formulas supply explicit homogeneous-shear fusion rules. The ordinary A-A pair is D_2D_1 S_y(2Q^(-1)), and scalar diagonal conjugation normalizes its factor 2 without Gaussian work. + +The common Q in the exceptional two-call formula is the existing central coordinate. Repeating the construction produces powers of that same Q, whose even powers are ordinary monomials. It does not create one fresh Gaussian rank per fused block. Likewise the free extraction on 56 ports does not diagonalize the other 56 A ports. A full caller must retain these exact facts when adding targets and costs. + +## T2. A Gaussian gauge on an ordinary role subspace + +Work over a field K of characteristic not 2 and let V be the data space. A scalar role projector P is an idempotent n by n matrix over K. It acts only on the n role indices, and therefore commutes with a common Gaussian Q on V. Define + + G_P(Q) = I + (Q−I)P. + +**Theorem.** G_P(Q) is invertible, with inverse G_P(Q^(-1)). It is ordinary-conjugate to Q on rank(P) role fields and identity on the other fields. It can be applied with rank(P) charged Gaussian jobs and explicit ordinary boundaries. + +**Proof.** The role space is the direct sum im(P) ⊕ ker(P): x=Px+(I−P)x, and the intersection is zero. Choose a basis in each summand and combine them into an ordinary scalar basis matrix U. In that basis P is diag(I_r,0), so G_P(Q) is diag(QI_r,I). The asserted inverse and construction follow. Alternatively, multiply the two displayed polynomial expressions in P and use P²=P. ∎ + +This is a changed gauge representation, not a claim that Q on an arbitrary role subspace is free. The basis maps and all Gaussian requests are specified. For a fixed finite template the ordinary maps have fixed coefficient data and actual finite field-pass cost. + +## T3. Coordinate-free rank tariff for an ordinary boundary + +Let M be an invertible scalar role matrix, with an input projector P and output projector R. Put + + F = G_R(Q) M G_P(Q)^(-1), + E = R M − M P. + +**Theorem.** There is an explicit all-input boundary construction for F using rank(E) charged Q or Q^(-1) jobs, ordinary linear operations and O(n) owned field buffers returned to zero. If E=0, F=M and no Gaussian correction is needed. + +**Proof.** Choose ordinary bases diagonalizing P on inputs and R on outputs. In those bases write M in four blocks M_00,M_01,M_10,M_11, according to the zero/one projector sectors. The physical boundary has ordinary diagonal blocks and crossed blocks M_01Q^(-1),M_10Q. The transformed E has crossed blocks −M_01 and M_10 and zero diagonal blocks; their input and output supports are disjoint, so + + rank(E)=rank(M_01)+rank(M_10). + +Rank-factor each crossed scalar block. Form its inner scalar combinations in owned temporary fields from one unchanged input snapshot, query each combination once, and distribute through the outer rank factor. Accumulate both direct and crossed terms in an owned output bank, swap it into the main bank after the last read, and clear the old bank and temporaries. This produces exactly F on every independent input and preserves unrelated fields. All Gaussian calls are counted. If E=0 the intertwining identity RM=MP gives G_R M=M G_P, hence F=M. ∎ + +This extends the row-bit boundary tariff to arbitrary scalar role subspaces. It does not extend it to an arbitrary matrix whose entries are noncommuting address operators. + +## T4. Composition and reciprocal-dual compatibility + +**Theorem.** For scalar M_1,M_2 and projectors P_0,P_1,P_2, define + + E_1=P_1M_1−M_1P_0, + E_2=P_2M_2−M_2P_1. + +Then the whole-boundary defect obeys + + P_2M_2M_1−M_2M_1P_0 = E_2M_1+M_2E_1, + +so its rank is at most rank(E_1)+rank(E_2). For invertible M, the reciprocal-dual defect is + + R^T M^(-T) − M^(-T) P^T + = M^(-T) (RM−MP)^T M^(-T), + +and has exactly the same rank. + +**Proof.** Expand the first identity; the two terms containing M_2P_1M_1 cancel. Rank is subadditive and cannot increase under multiplication. For the second identity transpose RM−MP and multiply by M^(-T) on both sides. Invertible left/right factors preserve rank. The dual gauge is G_(P^T)(Q^(-T)), so T3 applies to the actual reciprocal-dual operator at that same tariff. ∎ + +The global defect can have lower rank than the sum of local defects, including exact cancellation. This justifies compressing a completed ordinary boundary as one matrix rather than pricing a chosen ADD factorization as though its separate corrections could never share. + +## T5. Zero-correction endpoint gauges inside actual multiplicity spaces + +Assume a complete reference word satisfies + + L M R_0 = T, + +where L and R_0 are ordinary invertible scalar boundaries and T is the actual independent-output operator. Assume each target block of T commutes with the common Q. Suppose the internal entrance and exit gauges are G_A(Q) and G_B(Q), and there is a projector P such that + + R_0^(-1) A R_0 = P = L B L^(-1), + PT=TP. + +**Theorem.** The gauged interior has the exact same independent target under the same ordinary boundary maps: + + L G_B(Q) M G_A(Q)^(-1) R_0 = T. + +No Gaussian correction at those two boundaries is needed. + +**Proof.** The first projector identity gives G_A(Q)^(-1)R_0=R_0G_P(Q)^(-1). The second gives L G_B(Q)=G_P(Q)L. Therefore the left side is G_P(Q) T G_P(Q)^(-1). Since P commutes with T and the actual target blocks commute with the common Q, this equals T. All input fields remain arbitrary and independent because the complete endpoint is exactly the original T. ∎ + +For a scalar projector and T=diag(T_i), its commutator has entries P_ij(T_j−T_i). Thus a sufficient source of such projectors is an actual multiplicity block where the operators T_i are exactly identical. Equal Gaussian rank alone is not equality of these operators. Address-dependent normalizers must be included explicitly before applying the scalar statement. + +The theorem is a constructive admission criterion. This note has not supplied A,B satisfying it for a profitable complementary-provider schedule, so no free boundary in the current provider is asserted merely from the existence of this criterion. + +## T6. Sector-multiplexed Gaussian queries + +Suppose Q²=J and J is an ordinary involution, J²=I. Define E_+=(I+J)/2 and E_-=(I−J)/2. These are complementary ordinary projectors commuting with Q. Let A_+,A_- be scalar role matrices, with ranks r_+,r_-, and let + + A=A_+E_+ + A_-E_-. + +**Theorem.** The vector contribution A Q x can be computed using + + max(r_+,r_-) + +charged Gaussian jobs, explicit ordinary operations and cleared owned storage. + +**Proof.** Choose rank factorizations A_σ=U_σV_σ. Pad each inner dimension with zero rows/columns to r=max(r_+,r_-). Put + + U=U_+E_+ +U_-E_-, + V=V_+E_+ +V_-E_-. + +Scalar role matrices commute with the sector projectors. Since E_+E_-=0 and E_++E_-=I, UV=A. Form the r temporary fields Vx, apply Q once to each, and spread by U. Because Q commutes with all entries of U,V, the result is A Q x. Each temporary packs disjoint projected components, whose dimensions sum to the original data-space dimension; it does not pack two independent full data fields into one without accounting for information. Clear the owned storage after accumulation. ∎ + +For the actual lifted central Gaussian Q_f, Q_f² is the explicit translation that flips the selected bit in every digit column. E_± are therefore finite combinations of two ordinary monomial maps, uniformly in the lift width. The theorem has a concrete ordinary implementation and does not require a width-dependent spectral decomposition. + +## T7. A complete strict query-sharing example + +**Theorem.** Let Q be unitary with Q²=J as above. The invertible two-field map + + x ↦ x + (1/4)(Q+Q^(-1))x, + y ↦ y + (1/4)(Q−Q^(-1))y + +has an implementation using one Q job, plus ordinary operations and cleared owned storage. Its inverse norm is at most 2 in Euclidean norm. + +**Proof.** Since Q^(-1)=QJ, + + Q+Q^(-1)=2QE_+, + Q−Q^(-1)=2QE_-. + +Form z=E_+x+E_-y in an owned temporary, query Qz once, and return x+(1/2)E_+Qz and y+(1/2)E_-Qz. Orthogonality of the complementary sectors proves the two displayed outputs. The block-diagonal perturbation of identity has norm at most 1/2, so its Neumann inverse has norm at most 2. This norm argument establishes invertibility, not an unpriced inverse algorithm. The one-query forward construction is already explicit. ∎ + +The literal four-term implementation uses four Q or Q^(-1) calls; merely factoring the two odd expressions separately uses two. Sector multiplexing reduces that explicit tariff to one while preserving the full stated all-input map. These outputs are not declared canonical independent Gaussian targets. To improve κ this pattern must occur inside a completed provider whose original independent endpoint remains unchanged. + +## Exact contribution and κ status + +T1 supplies exact fusion normal forms for the actual typed B/C operators. T2–T5 provide role-subspace gauge admission and complete boundary tariffs, including duals. T6–T7 give a concrete way to share Gaussian queries across ordinary involution sectors, with a strict full-map example. + +A higher κ still requires a particular complete supplier whose actual paid profile is decreased by one of these replacements while preserving its independent target operators and uniform compiler premises. No such profitable B/C instance was established in this note. The constructions are positive, fully stated tools for that task; they are not a claim that the complex-only quotient headline has already satisfied the missing controller contract. + diff --git a/research/characteristic-gaussian-provider/whole-k.md b/research/characteristic-gaussian-provider/whole-k.md new file mode 100644 index 0000000000..7a5046c0fb --- /dev/null +++ b/research/characteristic-gaussian-provider/whole-k.md @@ -0,0 +1,176 @@ +# A smaller whole-K/E primitive from faithful radical completion + +Dugongue, with substantial OpenAI Codex assistance. This is informal mathematics based only on local DAG, linked Lean sources and retained proof notes. No numerical experiment, executable mathematical check, Lean run or external search is used. The result below constructs a changed local Gaussian provider with smaller ambient dimension and an explicit paid profile. It does not claim that its coded negative terminal has become an independent profitable Gaussian output. + +## Sources and exact interface + +The source `research/odd-e8-whole-k-native/PROOF.md` describes the whole-K geometry, actual native source/reader paths and real normalizers; the retained central-Q package reproduces those source statements. The retained `research/odd-e8-lowrank-e-native/E-PROOF.md` describes the ordinary mode recodings, the owner identity KC=−C, and the paid modal query. The retained `research/central-q-decoder/PROOF.md` and `E-COMPOSITION.md` give a complete paid terminal repair and its invoice. Credit THU New Media Research Center/PR5464, Ali Romman/PR5439, Dugongue, Dillon Amburgey, Cinna Davis, Ryanson Jonathan, Cao Yifan, Jacob Alan Sussman, Chris R/Clear1nsight, zxv and the native/Clifford/compiler/analytic lineage. + +The local TI/symplectic proof tree was inspected for its distinction between preserving a tensor's actual support and merely preserving pairwise contractions: `Mathproof/SymplecticTI.lean`, `Mathproof/LibraryBridges/Claude/TISpTransporterFibers.lean`, and `Mathproof/SymplecticTITOCIBalancedCoevaluation20260901.lean`. The compression below uses an injective isometry of a degenerate binary support, not an algorithm for tensor isomorphism and not an assertion that pairwise Gram equality alone preserves that support. The finite-form and Clifford tools are classical. The contribution here is their concrete provider reconstruction with its true changed outputs. + +All binary spaces below are over F₂. Write I_n for F₂^n with its usual dot product. Its characteristic vector is χ_n=(1,...,1), characterized by x·χ_n=x·x for every x. The complex Gaussian on a coordinate is C=((1+i)I+(1−i)X)/2. In a native block of ambient dimension n, R_n is the complete n-axis Gaussian, Q the first-axis Gaussian and D_n=R_n Q^(−1). + +## K1. The product cube has a small faithful support + +**Theorem.** Suppose one base eight-port cube has two four-port classes. Their spans have dimension three and Gram rank one, the two spans are orthogonal, and their intersection is their common two-dimensional radical. For an odd integer m, form the m-fold product cube and split its 8^m labels by total selector parity. Then each class spans a 3m-dimensional Gram-rank-one space; the two spaces are orthogonal and intersect in a common radical of dimension 3m−1. Their joint span has dimension 3m+1 and Gram rank two. + +**Proof.** In a base cube let the two spans be U,V and their intersection R. The source geometry gives dim R=2. Orthogonality implies R is contained in the radical of each three-space, hence equals each radical. Choose norm-one vectors u in U and v in V. They are orthogonal and + + U= direct_sum R, V= direct_sum R. + +The four distinct labels in a class are exactly its affine coset u+R or v+R, since each coset has four elements. Across m factors put R_all equal to the direct sum of the m two-dimensional radicals and w_j=u_j+v_j. A product label has the form + + u_1+...+u_m + sum_j σ_j w_j + r, + +with r in R_all. Two such labels of total selector parities ε and δ have dot product 1+ε+δ: in each factor their contribution is one for equal selector classes and zero for opposite classes, and m is odd. Thus equal-parity labels pair to one and opposite-parity labels pair to zero. + +Within a parity class the differences span R_all together with all even sums of the independent w_j. This has dimension 2m+(m−1)=3m−1. Adjoining one norm-one label gives dimension 3m and rank one. The two classes have that common difference space as their intersection; adjoining a label from the other class adds one dimension and one independent norm-one quotient direction. Their joint dimension is 3m+1 and rank two. ∎ + +For m=3 the dimensions are 9,9 and 10, with radical dimension 8. The base common-radical hypothesis can also be recovered from the retained three-factor support statement: a base affine-difference space has dimension at least three, since it contains the two-dimensional class radical and one interclass difference. The full three-factor affine-product span has dimension one plus the sum of those difference-space dimensions. The retained dimension ten forces each to have dimension three. Therefore the two class radicals coincide and have dimension two. This proves the support structure used here instead of treating the global 25-dimensional realization as a minimal one. + +## K2. An explicit faithful embedding into an even ambient space + +**Theorem.** A binary space S with an orthogonal decomposition + + S= direct_sum R, + u·u=v·v=1, u·v=0, dim R=r, + +where R is its radical, embeds injectively and isometrically into I_(2r+2). Its norm-one labels are not the ambient characteristic vector. For the product cube of K1 this gives ambient dimension 6m; at m=3 it gives I_18. + +**Proof.** Map u to e_1, v to e_2, and a basis r_1,...,r_r of R to + + e_3+e_4, e_5+e_6, ..., e_(2r+1)+e_(2r+2). + +These images are linearly independent. They have exactly the required pairings: the first two vectors have Gram I₂, every paired sum has norm zero, and all pairings between distinct displayed generators vanish. The linear map is therefore an injective isometry. The ambient characteristic vector has norm 2r+2=0 in F₂, while every port label has norm one, so no port equals it. Apply r=3m−1 from K1. ∎ + +The absence of a norm-one ambient characteristic vector is helpful here, not an obstruction. The native source requires its port labels to be norm one and different from χ_n; it does not require the distinguished central coordinate to equal χ_n. Adding one extra coordinate gives the conservative odd ambient 6m+1, but that addition is unnecessary for this reconstruction. + +The embedding is faithful on the whole degenerate span. Simply identifying every label of a parity class would preserve its pairwise Gram values but collapse its radical and change the physical code more drastically; K2 does not make that identification. + +## K3. Dimension-free canonical frames and their complete paths + +**Theorem.** In ambient I_n, the source's canonical-frame construction works for every subspace S, including a degenerate one. Its image of the computational Z Lagrangian is + + L_S={(x,z): x belongs to S and z+x belongs to S-perp}. + +For S contained in T, the Gaussian rank of the transition is dim T−dim S. Consequently, if a source parity class has dimension k and an opposite reader label q is norm one and orthogonal to that class, the four source/reader paths have ranks + + k−1, k, n−1−k, n−k. (1) + +**Proof.** Choose any full binary basis G whose first dim S columns span S, write A=G^(−1), and use the source's computational monomial K_G=D_J P_G with J=I+A^T A. The frame is K_G C_[0,dim S) K_G^(−1). Its symplectic image is the displayed L_S by the direct calculation in the source: the X component lies in S and adding it to the Z component is orthogonal to S. This calculation has no condition that n be odd or that the restriction of the dot product to S be nondegenerate. + +If S is contained in T, a vector in L_S intersect L_T has x in S and z+x in T-perp. Thus the intersection dimension is dim S+n−dim T. The Gaussian rank is n minus that dimension, giving dim T−dim S. A full flag extending S inside T gives the literal partial-Gaussian implementation, with computational monomial chart normalizers. + +For (1), the source line has dimension one and is contained in the k-dimensional common class, giving k−1. The reader starts at ground and reaches that common class with rank k. Since q is orthogonal to the class, q-perp of dimension n−1 contains it, giving n−1−k for the reader exit. The source exits to the full n-space with rank n−k. These are the four actually executed Gaussian children, not a rank of a scalar coupling matrix substituted for a child. ∎ + +For n=6m and k=3m, (1) becomes + + 3m−1, 3m, 3m−1, 3m. + +At m=3 it is 8,9,8,9. The original ambient-25 construction instead has 8,9,15,16. + +## K4. The ordinary wrappers and their absolute phases survive + +**Theorem.** For every norm-one noncharacteristic port q in I_n there is an explicitly constructed real signed monomial L_q sending the central line to the q line and commuting exactly with R_n. Therefore the native line-service input and reader-output wrappers of the source can be constructed in ambient 6m, with no extra unpriced Gaussian child. The all-width statement includes absolute phase. + +**Proof.** An even-vector orthogonal transvection is T_w(x)=x+(x·w)w, with w·w=0. If the central coordinate of q is zero, T_(e_0+q) sends e_0 to q. If it is one, either q=e_0 or choose j>0 with q_j=0, possible because q is not characteristic; then T_(e_j+q) T_(e_0+e_j) sends e_0 to q. This gives a binary orthogonal matrix G_q. + +For row j of G_q put z_j=(wt(row_j)−1)/2 modulo two and let L_q=Z_z P_Gq. Orthogonality implies the finite weight identity + + wt(G_q^(−1)x)−wt(x)=2 z·x modulo four. + +Indeed quadratic cross terms cancel, leaving precisely the displayed linear coefficients. Hence the real signed monomial L_q commutes with R_n exactly, by substituting the coefficients of the tensor Gaussian on each basis label. + +The literal source wrappers are Φ_[q] L_q Q^(−1) on the source input, L_q on the reader input, L_q^(−1) on the source output, and D_n L_q^(−1) Φ_(q-perp)^(−1) on the reader output. Their computational monomiality follows from their identical computational Lagrangian images. No scalar phase is discarded: the source's dimension-free anchor is + + R_n^(−1)|0> = ((1−i)/2)^n sum_x i^(wt(x)) |x>, + R_n K_G R_n^(−1)|0> = (−i)^(wt(d)) |d>, + +with d_j=floor(wt(G^(−1)e_j)/2) modulo two. The same finite weight calculation proves that anchor for every n. It determines the phase of each full monomial word, and the remaining line/hyperplane wrappers use the source's two-one-axis low-rank defining expressions. + +Finally tensor the one-copy operator identities over f positive digit columns. Scalar owner coefficients occur once per field; monomial phases sum over columns and the fixed scalar phase is raised to f. No tensor power of a sum is asserted. This proves the all-width endpoint with absolute phase. ∎ + +The claim is a mathematical construction of the wrappers. No new source file, program replay or formal proof check is being claimed. + +## K5. A compressed complete local whole-K service + +**Theorem.** For each odd m, let K_owner be a real symmetric involution on the 8^m owners whose nonzero entries join opposite total-selector parity classes. Use the faithful K2 geometry and K4 wrappers. Then the actual arbitrary-input two-bank endpoint is + + F_K = diag(D_n,D_n) L(C_K Q^(−1)), + (C_K)_ij=(K_owner)_ij L_i^(−1)L_j, + n=6m, + +with no independent helper input and with paid profile, per owner, + + 2 H_(3m−1) + 2 H_(3m). + +C_K is a real symmetric involution and commutes R_n. + +**Proof.** At the common parity-class frame, execute the ordinary scalar ADDs Y_i +=(K_owner)_ij X_j for all nonzero coefficients. Opposite classes have identical required contact frames by K3. Every source is read before its value is changed; every reader retains its arbitrary old value. The geometric endpoint is the same source/reader formula as in the retained proof, now in ambient n. + +After the K4 wrappers the source output is D_n x_i because L_i commutes R_n. The reader output is + + D_n y_i + sum_j (K_owner)_ij D_n L_i^(−1)L_j Q^(−1)x_j, + +which is exactly the claimed triangular map. It is invertible on arbitrary independent x,y. K3 supplies the four paid paths. Writing L=diag(L_i), one has C_K=L^(−1)(K_owner tensor I)L, so its square, symmetry and commutation with common R_n follow from the stated real orthogonality and K_owner identities. No paid query input is credited as an independent helper output. ∎ + +The actual odd-majority owner code has the required parity support. To see this generally, define K_owner as the odd-majority function of m commuting symmetric involutions K_1,...,K_m. Its eigenvalues are ±1, so it is a symmetric involution. Replacing all K_j by −K_j changes the majority sign, so its multilinear Fourier expansion contains only odd products. Each K_j flips one selector parity; every odd product therefore flips total parity. This supplies the hypothesis. At m=3 it is precisely (K_1+K_2+K_3−K_1K_2K_3)/2 from the source. + +For m=3 every new product-cube word thus has paid profile + + {8:2v_cube, 9:2v_cube}, v_cube=512, + +rather than {8:v_cube,9:v_cube,15:v_cube,16:v_cube}. This is a changed local service: D_n has rank 17 rather than 24, and its physical C_K depends on the new normalizers. It is not the old rank-24 endpoint implemented for the smaller price. + +## K6. Cross-cube ordinary modes can be rebuilt consistently + +**Theorem.** All product cubes can be reconstructed in one common I_18 for m=3, while retaining separate independent owner fields. Let K_owner be their block-diagonal scalar code and let C_owner be the retained ordinary owner matrix with K_owner C_owner=−C_owner and spectrum {0,98,−1078}. Put + + C_new=L^(−1)(C_owner tensor I)L, + E_new=C_K (I−C_new)^(−1). + +Then all algebraic and modal-query steps of the source E adapter remain valid on this newly constructed physical service. Each corrected orientation has paid profile + + {8:2v, 9:2v, 18:h}, + v=112^3, h=20480. + +**Proof.** Cubes act on disjoint owners, so their labels may be separately embedded by K2 into the same address ambient. No two independent payloads are identified even if some newly assigned labels coincide across cubes. Each L_i commutes the same R_18. Consequently C_new, C_K and E_new commute R_18, and their scalar similarity preserves all owner spectral and inverse identities. + +Acquire the nonzero C_owner modal coordinates after the ordinary L_i normalizations; apply one R_18 to each of the h actual acquired modes; scatter by the prescribed eigenvalues; undo the ordinary mode and L_i changes. This is exactly R_18 C_new applied to the original source bank. The modes are obtained from actual original inputs and are not productive helpers. The source E algebra then corrects the local shear coefficient and gives the stated forward/backward forms. The inverse-contact construction and linear transposition use the same literal source/reader incidence from K5, so their all-input endpoint and paid rank profile are unchanged by replacing the ambient dimension and actual normalizers consistently. + +The K5 children give two copies each of ranks 8 and 9 per owner. Each modal query is now rank 18 and there are h of them. This proves the profile. ∎ + +The cross-cube owner matrix need not preserve the original geometric pairings between cubes. It is an ordinary scalar recoding, and the argument uses its new physical similarity explicitly. This is why the result is valid as a newly constructed provider and why it is not a free coordinate compression of the pinned 25-axis E operator. Claiming the latter would require a common spectator and a conjugacy preserving the old central-Q itinerary, which is not supplied here. + +## K7. Complete three-stage outputs and their honest invoice + +**Theorem.** Use three fresh 17-axis blocks with one common central Q, and the K6 corrected E modules. Let P=E_3 E_2^(−1)E_1. The actual two terminal operators, after the ordinary signed role swap, are + + Q^(−1) P D_123, P D_123 Q. + +The ordinary inverse P^(−1) closes the positive terminal. A complete negative repair by Q, P^(−1), Q gives independent rank-52 Gaussian outputs on both banks, with exact paid profile + + targets: 2v H_52, + paid: 3v(2 H_8+2 H_9) + 3h H_18 + 2v H_1. (2) + +**Proof.** The source three-stage operator derivation uses only the actual corrected E interfaces, commutation with each stage's R=DQ, and identity action on other fresh blocks. Those conditions hold in K6, so its chronological product gives exactly the displayed terminals. Applying Q to the negative terminal gives P D_123; applying the ordinary reversed inverse word gives D_123; applying Q gives D_123 Q. The positive terminal needs only P^(−1). Since there are three fresh rank-17 blocks and one central line, both final Gaussian ranks are 3·17+1=52. + +There are three corrected local calls, each with the K6 profile, and two additional central queries on the negative bank. All other inverse recodings are ordinary, with their finite coefficients, owned storage and error amplification still charged. This proves every term of (2). ∎ + +Its rank-mass surplus is −54h: the target mass is 104v, the native local mass is 102v, and the two central repairs add 2v, leaving the modal 54h. Moreover this particular complete recipe has negative target-minus-paid concave moment for every 00 the compressed corrected local paid moment is strictly smaller than the old one: + + v[(15^t−8^t)+(16^t−9^t)] + h(25^t−18^t) > 0. + +This is immediate from strict increase of x^t. The endpoint ambient and physical code have also changed, so this local inequality cannot be substituted into an old whole-provider invoice with its rank-24 or rank-73 targets untouched. K7 rebuilds the corresponding new invoice exactly. + +The actionable provider direction is now sharper: construct a cheaper complete consumer for the new 18-axis E code, or exploit its retained radical/owner structure to close a different independent-output word. The conditional screen that simply grants both negative and positive rank-52 outputs is not a theorem. Neither Gram compression, total symmetry nor nonzero support supplies that missing decoder by itself. + +All matrices, modes, phases and storage discussed here have explicit mathematical definitions. The note supplies no new numerical κ, native executable, formal-kernel acceptance or claim of historical priority. Its useful sufficient construction is a full local K/E provider reconstruction and paid all-input completion at the new dimension. + +