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| 1 | +# Terminal decoder admission: routing erases almost all summary overlap |
| 2 | + |
| 3 | +2026-10-09. Douglas Colkitt, with OpenAI Codex assistance. Apache-2.0. |
| 4 | + |
| 5 | +**No terminal decoder is admitted, and κ is unchanged.** The selected complex |
| 6 | +cover forces this four-summary factorization to retain at least `119/120` |
| 7 | +of its uncompressed summary dimension. A common-frame terminal implementation |
| 8 | +also consumes full rank capacity, even with free internal mixing. Neither |
| 9 | +finding excludes every distributed realization of the scalar decoder. |
| 10 | + |
| 11 | +This completes the bounded admission pass from the |
| 12 | +[design reset](../mechanism-design-reset/REPORT.md). No frame search or |
| 13 | +production-verifier change was launched. |
| 14 | + |
| 15 | +## Selected inputs and routing |
| 16 | + |
| 17 | +The audit reads frozen `selected/complex/graph.json.gz`. It checks all 1,320 |
| 18 | +ports at `h=22`, selecting one coordinate from each of three coordinate |
| 19 | +pairs. It reconstructs the signed ancestors of all 22 center roots and |
| 20 | +checks that they produce the incidence sums `sum_(S contains i) x_S`. |
| 21 | +Every scatter coefficient is checked against `1/2·1_(i in T)−1/6`. |
| 22 | +Thus the central map is `C_TS=(|T intersect S|-1)/2`, factored through |
| 23 | +the full-rank, 22-row incidence matrix V. |
| 24 | + |
| 25 | +The remaining side map is `I-C`, including both the paired-cube side |
| 26 | +producer and original-source involution. It is not just the local Hadamard |
| 27 | +block. See the [retained construction](../../notes/paired-cube-construction.tex). |
| 28 | +The receipt pins the graph and cover/frame notes. It does not assume that |
| 29 | +deleting center roots preserves old dirty-response corrections or rerun |
| 30 | +the entire physical word. |
| 31 | + |
| 32 | +Write `G=O(66,2)`, `g=|G|`. The retained cover places data port `(a,S)` at |
| 33 | +vertices `aR_12,S`, `a`, `aR_23,S` in the three stages. Its exchanges satisfy |
| 34 | + |
| 35 | +\[ |
| 36 | + R_{12,S}A=B\oplus\langle q_S\rangle,\qquad |
| 37 | + R_{23,S}A=C\oplus\langle q_S\rangle. |
| 38 | +\] |
| 39 | + |
| 40 | +Distinct ports give distinct exchanges within either family: the image of |
| 41 | +A determines the source line. This holds for every permitted exchange-basis |
| 42 | +choice. The script constructs one canonical choice for all 2,640 maps and |
| 43 | +checks orthogonality, involution, fixed source, active image and fixed padding. |
| 44 | + |
| 45 | +Let `A₂=I_g tensor V`. The other summary maps are this block incidence map |
| 46 | +after the respective data-column permutations. Auxiliary routing only |
| 47 | +permutes rows and does not change row spaces. Each has rank `gh`. |
| 48 | +The specified decoder needs `A₂x,A₃x,A₁y,A₂y`, with joint dimension |
| 49 | + |
| 50 | +\[ |
| 51 | + r_* = \operatorname{rank}\begin{pmatrix}A_2\\A_3\end{pmatrix} |
| 52 | + +\operatorname{rank}\begin{pmatrix}A_1\\A_2\end{pmatrix}. |
| 53 | +\] |
| 54 | + |
| 55 | +This concerns these input summaries, not the minimum workspace of every |
| 56 | +possible decoder. Output-side cancellations, different factorizations and |
| 57 | +sequential recomputation are not ruled out by this dimension count. |
| 58 | + |
| 59 | +## A projection trace bound on the full cover |
| 60 | + |
| 61 | +Use ordinary rational/real orthogonal projections onto scalar row spaces, |
| 62 | +not binary address frames. Set `L=V^T(VV^T)^−1 V`. The actual Gram matrix |
| 63 | +has diagonal 180, partner-coordinate entries zero, and other entries 18. |
| 64 | +Exact inversion and all port checks give `L_SS=h/v=1/60`. |
| 65 | + |
| 66 | +Let P_j project onto the row space of A_j. Their intersection dimension |
| 67 | +is at most `tr(P_i P_j)`: the trace contributes one on the intersection |
| 68 | +and nonnegative terms on its orthogonal complement. |
| 69 | + |
| 70 | +The untwisted projector connects `(a,S)` to `(a,T)`. A neighboring stage |
| 71 | +connects these same positions only when `aR_j,S=aR_j,T`. Distinct routes |
| 72 | +therefore leave only `S=T` in the trace expansion. Consequently |
| 73 | + |
| 74 | +\[ |
| 75 | + \operatorname{tr}(P_2P_j)=g\sum_S L_{SS}^2 |
| 76 | + =g\frac{h^2}{v}=\frac{11g}{30}\quad(j=1,3), |
| 77 | +\] |
| 78 | + |
| 79 | +and |
| 80 | + |
| 81 | +\[ |
| 82 | + r_*\ge g\left(4h-\frac{2h^2}{v}\right) |
| 83 | + =\frac{1309g}{15}\approx87.2667g. |
| 84 | +\] |
| 85 | + |
| 86 | +This is a full regular-cover identity, not a sampled group calculation. |
| 87 | + |
| 88 | +| Summary accounting | Dimension | |
| 89 | +|---|---:| |
| 90 | +| Identical, aligned stage maps | `44g` | |
| 91 | +| Actual routing: lower bound for these summaries | `87.2667g` | |
| 92 | +| Four separate banks | `88g` | |
| 93 | + |
| 94 | +At most `1/120` of the separate dimension can disappear through row-space |
| 95 | +overlap. The lower bound is `119/60` times the aligned budget. It exceeds |
| 96 | +the `66g` stage-indexed center values in three separate invocations, but |
| 97 | +that is **not** a recursive cost comparison: old stock is reused, and |
| 98 | +widths, chronology and restoration differ. |
| 99 | + |
| 100 | +A dense rational control over the cyclic group of order three checks the |
| 101 | +trace formula independently. A collided-routing control gives trace six |
| 102 | +rather than four, rejecting use of the diagonal-only formula without its |
| 103 | +distinct-route premise. |
| 104 | + |
| 105 | +## Common-frame decoding exhausts rank capacity |
| 106 | + |
| 107 | +Keep the old data cuts and full-volume child rule. Align all updated data |
| 108 | +to common exact Clifford frames for terminal decoding, allowing different |
| 109 | +common input/output frames and granting internal mixing for free. Retained |
| 110 | +controls with converted copies are allowed. |
| 111 | + |
| 112 | +After the old complement tails, data gauges are `X:F`, `Y:FT_(aq_S)^−1`, |
| 113 | +with the actual source Pauli/sign normalizations retained. For norm-one |
| 114 | +sources the Y preimage frames are `L_((aq_S)^perp)`. Distinct normals have |
| 115 | +Lagrangian distance two. The |
| 116 | +[arbitrary-frame metric](../../notes/general-clifford-frames.tex) gives, |
| 117 | +for any common Clifford frame M, |
| 118 | + |
| 119 | +\[ |
| 120 | + d(L_{q^\perp},M)+d(M,L_{r^\perp})\ge2\quad(q\ne r). |
| 121 | +\] |
| 122 | + |
| 123 | +At each fixed a there are v distinct port normals. Summing over unordered |
| 124 | +pairs gives total one-way distance at least v. Over the cover this is |
| 125 | +`N=gv`; returning the updated Y outputs costs another N. The same bound |
| 126 | +allows a different common frame for every fixed-a group. Normal-specific |
| 127 | +frames with paid interactions instead constitute a distributed model |
| 128 | +outside this screen. |
| 129 | + |
| 130 | +The old prefix costs at least `2N(m−1)` on data: source lines reach the |
| 131 | +full frame and zero-frame targets reach complementary hyperplanes. Thus |
| 132 | + |
| 133 | +\[ |
| 134 | + 2N(m-1)+2N=2Nm. |
| 135 | +\] |
| 136 | + |
| 137 | +Per group element at m=66 this is 171,600 prefix rank plus 2,640 alignment |
| 138 | +rank, equal to data capacity 174,240. Under the retained restored |
| 139 | +full-transform endpoint, each independent dirty role adds at least m |
| 140 | +along its physical chain. Changing their count does not rescue this model. |
| 141 | + |
| 142 | +For `0<s<1`, `(t/m)^(1-s)>=t/m`. Rank at least capacity therefore prevents |
| 143 | +a contracting moment. This grants zero internal decoder cost, free X |
| 144 | +alignment and no extra fallback or precision cost. Pauli/phase adapters |
| 145 | +cannot improve this lower bound; no equality of symplectic labels is used |
| 146 | +to claim an exact implementation. |
| 147 | + |
| 148 | +The small control tests all 67 subspace frames in binary dimension four |
| 149 | +against eight distinct norm-one caps; the best one-way sum is eight. |
| 150 | +The all-frame statement follows from the metric proof, not the catalog. |
| 151 | + |
| 152 | +## Disposition and scope |
| 153 | + |
| 154 | +Stage summaries scarcely overlap, and common-frame terminal decoding loses |
| 155 | +even with free internal work. No new complete child histogram or better |
| 156 | +moment results. Do not start a large common-frame or summary-compression |
| 157 | +search on this factorization. |
| 158 | + |
| 159 | +The scalar decoder remains valid. A distributed implementation might |
| 160 | +interleave summaries with differently framed data, change cuts, or avoid |
| 161 | +these summaries. **Its cost remains unresolved.** The routing bound alone |
| 162 | +does not exclude it, and the common-frame bound must not be generalized |
| 163 | +to it. This is not a proof that all coupled decoders are impossible. |
| 164 | + |
| 165 | +A successor needs a concrete distributed word identifying paid work that |
| 166 | +is shared or removed. Another optimistic rank factorization is insufficient. |
| 167 | +Stop this bounded round with the conditional witness unchanged at |
| 168 | +`330942774629799/500000000000000000`. |
| 169 | + |
| 170 | +## Reproduction and attribution |
| 171 | + |
| 172 | +```sh |
| 173 | +python3 -B research/terminal-decoder-admission/audit.py --check research/terminal-decoder-admission/certificate.json |
| 174 | +python3 -B -m unittest discover -s research/terminal-decoder-admission -p 'test_*.py' |
| 175 | +make selected-record-check |
| 176 | +``` |
| 177 | + |
| 178 | +The graph is Dugongue #186 on chafreaky #181 and eumemic #168. The paired-cube |
| 179 | +construction and orthogonal cover retain icekylinx's and cumulative community |
| 180 | +attribution, including an664's completed-core sharing. These diagnostics |
| 181 | +neither modify the frozen supplier nor establish its inherited all-size |
| 182 | +interfaces independently. |
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