Add exact complex rank-16 block decoder screen

research/complex-e8-pair-block-climb/README.md

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+# An exact complex decoder climbs from two E8 roots to rank 16
+
+**Scoped partial result.** The exact tensor decoder and support/COPY ledger are
+verified. Its copy-only profile clears the selected model's five-percent
+ceiling screen, while one explicit frame-slot tariff sensitivity falls below
+the selected endpoint. No physical tariff, selected supplier, new κ, or
+frontier change is established.
+
+## Question and result
+
+Can a small complex decoder start from actual E8 roots, climb through block
+ranks 2, 4, 8 and 16, and fit the selected call's full copy/frame ledger?
+
+The exact identity exists, and its support-pairing matrix has full rank at each
+level. The rank-16 block has 16 full-support terms of copy rank 16, so its COPY
+mass is 256. The selected model has 896 live families over nine stages. The
+rank-16 construction therefore uses 56 blocks per nine-stage replica, or an
+average `56/9` blocks per call. Keeping the selected eight rank-8 local terms,
+the per-call profile is `{16:896/9, 8:8}` with local COPY mass `14912/9`.
+
+Under the pinned parent relaxation, that copy-only profile has ceiling bracket
+`[1942057021/250000000000, 1553645617/200000000000]`, whose upper endpoint is
+about `0.007768228085`. This exceeds the five-percent target
+`0.00177455681454345`. The result is only an ideal profile screen: product
+levels 2–4 use a new label form, and the frame ledger has no established tariff
+in the parent model. Charging one rank-1 unit for each listed per-block frame,
+embedding, restoration-read and scalar-add slot gives `12824/3` extra units per
+call and a sensitivity ceiling near `0.001658774411`, below the selected
+`0.001690054109089` endpoint. The physical tariff and a selected-code embedding
+remain open.
+
+## Exact decoder identity
+
+Take the two E8 roots
+
+```text
+r0 = (1,1,0,0,0,0,0,0)
+r1 = (1,0,1,0,0,0,0,0)
+```
+
+Both have norm squared 2 and dot product 1, so their Gram matrix is
+
+```text
+G0 = [[2,1], [1,2]].
+```
+
+Over `Q(i)`, with **ordinary transpose**, set
+
+```text
+D0 = [[1, 1], [1, i]]
+T0 = [[3-i, 1+i], [3+i, -1-i]].
+```
+
+Direct multiplication gives `D0 T0^T = [[4,2],[2,4]] = 2 G0`. Every entry of
+both frames is nonzero. The two E8 support labels pair by
+`(<r_i,r_j> + 1) mod 2`, giving `B0 = I2`; each of the two terms therefore has
+support-pairing rank 2 and the base copy mass is 4. Using conjugate transpose
+does not give this identity (checked as a negative control).
+
+For `s=1,2,3,4`, let `n=2^s`, `Rs=R0^tensor_s`,
+`Gs=G0^tensor_s`, `Ds=2^(1-s) D0^tensor_s`, and `Ts=T0^tensor_s`. Then
+
+```text
+Ds Ts^T = 2 Gs = 2 Rs Rs^T.
+```
+
+The verifier checks every matrix entry. The block-climbing frame itinerary is
+the repeated tensor step `s -> s+1`: the product-root frame has `n` rows in
+ambient dimension `8^s`; the decoder and reader frames have shape `n x n`;
+their ranks are `n`; and their exact inverses are checked. The product-root
+embedding has `n²` nonzero coefficients. The left inverse `Ls=L0^tensor_s`,
+where `L0` selects coordinates 2 and 3 of each base root, satisfies
+`Ls Rs^T=I_n`; it uses `n` coordinate reads and restores every coefficient
+vector in the product-root span. This does not restore arbitrary ambient state
+outside that span.
+
+The binary support pairing at level `s` is `Bs=(Gs + J_n) mod 2`, where `J_n`
+is the all-ones matrix. Modulo 2, `G0=[[0,1],[1,0]]`, so `Gs` is a permutation
+matrix `Ps`. Since `n` is even, `Bs=Ps(I+J_n)` is nonsingular: over F2,
+`det(I+J_n)=1+n=1`. The verifier independently computes all four ranks. For
+`s>=2`, these are product roots in `R^(8^s)` and a new `n`-bit label form
+`Bs`; they are not the selected 9-bit E8 port code.
+
+## Full block and per-call ledger
+
+| Tensor factors | Product lines | Terms | COPY rank per term | COPY mass | Forward D/T frame coefficients | Root embedding coefficients | Root-span restoration reads |
+|---:|---:|---:|---:|---:|---:|---:|---:|
+| 1 | 2 | 2 | 2 | 4 | 8 | 4 | 2 |
+| 2 | 4 | 4 | 4 | 16 | 32 | 16 | 4 |
+| 3 | 8 | 8 | 8 | 64 | 128 | 64 | 8 |
+| 4 | 16 | 16 | 16 | 256 | 512 | 256 | 16 |
+
+At level 4, both forward `16 x 16` frames are dense. Their four-stage `2 x 2`
+tensor application takes 272 constant-multiplication slots (including the
+`1/8` normalization on D) and 128 additions for the two input frames. If a
+consumer needs vector outputs returned to the original coefficient frames,
+the two inverse frames are also dense, another 512 coefficient slots per
+block. The scalar bilinear decoder returns a scalar and does not itself need
+that output-vector restoration. Each block also uses 16 rank-16 bilinear child
+products and 15 scalar accumulation additions.
+
+| Per-call quantity | Candidate profile | Selected model |
+|---|---:|---:|
+| Live families | `896/9` | `896/9` |
+| Average rank-16 blocks / children | `56/9` / `896/9` | — |
+| Retained rank-8 local terms | 8 | 8 |
+| Decoder COPY mass | `14336/9` | — |
+| Total local profile mass including retained rank-8 terms | `14912/9` | 5,586 |
+| Full local-plus-centre coverage mass available | — | 7,232 |
+
+The ratio of candidate to selected local mass is `7456/25137`. The 7,232
+coverage allowance follows the pinned model's `72*(896/9)+64` identity; its
+separate centre charge is retained. Under the parent ideal relaxation the
+candidate profile's ceiling is above the five-percent target. The ceiling
+assumes fractional idle counts and unconstrained repacking; it does not prove a
+physically realizable supplier.
+
+The level-4 per-call frame ledger over the average `56/9` blocks is:
+
+| Item | Slots or coefficients per call |
+|---|---:|
+| Forward D/T dense coefficients | `28672/9` |
+| Inverse D/T coefficients if vector outputs are required | `28672/9` |
+| Staged D/T constant multiplications | `15232/9` |
+| Staged D/T additions | `7168/9` |
+| Product-root embedding coefficients | `14336/9` |
+| Root-span restoration reads | `896/9` |
+| Scalar output accumulation additions | `280/3` |
+
+For a tariff sensitivity only, charge one rank-1 unit per staged multiply,
+addition, embedding coefficient, restoration read and scalar accumulation
+addition. This is `687` units per block and `12824/3` units per call. The parent
+profile can absorb at most 3,919 extra integer rank-1 units while clearing the
+five-percent target; 3,920 fails. The sensitivity ceiling is therefore below
+the selected endpoint. This unit tariff is not asserted to be the physical
+cost, and the dense inverse is excluded unless vector restoration is needed.
+
+## Relation to retained work and credit
+
+The selected baseline is pinned through `certificates/selected-result.json`,
+its manifest `research/community-frontier1094-audit/SELECTED-SOURCE.json`, its
+assembly `research/community-frontier1094-audit/ASSEMBLY.json`, and the exact
+review `docs/research/community-batch1099-review.md`. The selected assembly
+SHA256 is `1122cb3a575f8c788345fc1bf4c108ac7c3250fb81a5988671f5eb04b3ab3045`.
+The selected conditional κ is `0.001672660197076918`; the selected local-word
+ceiling is `0.001690054109089`. The first-step rank-16 ideal is a 2.31x
+relaxation and omits realization costs.
+
+Overlap checked live Start-here discussion #1163, handoff #1166, and relevant
+PRs. PR #3039 has a charged-group block-climb obstruction, while #3066 corrects
+its mixed-profile inference. PR #1633 bounds excess rank for all-pairs E8
+decoders and gives a ten-term normal form; #3566 binds a different masked P112
+tetrad decoder to selected ports. PRs #3701, #3805, #3811 and #3874 remain
+rank-cap and ideal-budget screens. This package contributes the explicit
+complex factorization on an actual E8 root pair, its four-level tensor
+support/COPY ledger, and an exact frame-slot tariff sensitivity. It does not
+duplicate the fixed-alphabet effective-rank result assigned to macbook-05 or
+claim the product roots embed in the selected ports. Credit prior proofs and
+source authors listed in `docs/research/complex-decoder-scope-map.md`,
+including PR717, PR968 and PR1077.
+
+## Reproduce
+
+From the repository root, Python 3.12 and the standard library:
+
+```sh
+python3 -I -B research/complex-e8-pair-block-climb/verify.py
+python3 scripts/research_package.py --package research/complex-e8-pair-block-climb
+```
+
+Expected direct replay:
+`PASS_COMPLEX_E8_PAIR_BLOCK_CLIMB levels=4 rank16_block_copy_mass=256 average_profile_mass=14912/9 ideal_a=0.007768228085 frame_proxy_a=0.001658774411`.
+The package replay checks the source pins and runs the explicit verifier command
+recorded in `VERIFY.json`.
+
+## Open next questions
+
+After the companion fixed-alphabet effective-rank result is available, does
+its surviving support class embed this factorization into the selected 9-bit,
+112-port pairing code, and what established physical tariff applies to the
+complete frame ledger? Until then, stop; a copy-only ceiling pass does not
+justify another construction round under the family-level expansion gate.
+
+Prepared with substantial OpenAI Codex assistance by homelab-05. No independent
+human review or formal proof-assistant verification is claimed. New material
+is Apache-2.0.