Prove a third-moment QQ16 transcript bound

research/qq16-third-moment-capacity/PROOF.md

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+# A third-moment lower bound for the QQ16 transcript
+
+Fix the conserved F0/B_1_2 maps and ordinary tail of Ali Romman's
+[PR1969](https://github.com/CrocSwap/integer-mult-bounds/pull/1969), retained
+in `research/joint-query`. Let G be a finite group of actual owner routes,
+acting regularly, and write N=|G|. Use cover-dependent fields constant in
+every payload block. Payload translations then act trivially, but the
+owner action remains regular. No nonfaithful quotient identifies routes.
+
+Under the inherited erased-family observation contract, an additional
+read-only rational-linear record of rank at most d leaves at least
+
+```text
+max(0, ceil(176N/13) - d)
+```
+
+original selected A/T dimensions invisible. In particular, thirteen
+rational scalars per owner leave at least ceil(7N/13) invisible dimensions.
+Fourteen is necessary, not proved sufficient or a sharp minimum.
+
+The two-moment method is Ross Richardson's PR2104. The joint A/T defect
+identity is commdata2338's PR2153. This contribution supplies the third
+moment and the spectral polynomial that strengthen their bound.
+
+## 1. Exact route premise
+
+Write B_s=R_s iota_U for s=0,1,2. The ordinary set U has 96 ports.
+For each p in U let r_p be its eight-component integer query column and
+let g_s(p) be its owner route. The frozen packet gives
+
+```text
+g_0(p)=1,       ||r_p||^2=4,
+sum_p r_p r_p^T=48 I_8.
+```
+
+Each g_s(p), s=1,2, swaps an ordered orthonormal eight-frame C_p in the
+first nine binary axes with eight external axes E_s. The external axis
+sets are disjoint: E_1 uses axes 9 through 16, E_2 uses 17 through 24.
+Each route fixes the unused external block and the remaining 48 axes.
+The 96 ordinary ordered frames C_p are distinct.
+
+These are finite source facts, checked on the literal F1/F2 alignment
+maps and rebound to the current and future alignments in B_1_2. The
+verifier checks both the complete ordered-product sets and an independent
+chart-swap reconstruction. It does not execute the native local words.
+
+The swap description proves
+
+```text
+{g_1(p)g_1(q): p,q in U} intersect
+{g_2(r)g_2(t): r,t in U} = {1}.
+```
+
+Indeed, a stage-2 product fixes E_1 pointwise. If a stage-1 product also
+fixes E_1, then g_1(q)e_i=C_q,i must equal g_1(p)^-1 e_i=C_p,i for every
+external basis vector e_i. Distinctness of the ordered frames forces p=q.
+The same argument gives r=t. Thus identity products have only diagonal
+port pairs. The independent full census gives 9,216 ordered pairs and
+9,121 distinct products per stage, with exactly one common product.
+
+The relative routes g_s(p)g_t(p)^-1 are also distinct over the 96 ports
+for each s<t. The executable checks all three pairs. For the stage-1/2
+pair this also follows from the product intersection above.
+
+## 2. The third moment
+
+For the regular owner action Q_g, tr(Q_g)=N for g=1 and zero otherwise.
+Consequently B_s B_s*=48 I_(8N). Set
+
+```text
+P_s = B_s*B_s/48,          T = P_0+P_1+P_2.
+```
+
+Each P_s is an orthogonal projection of rank 8N. The two-moment source
+calculation gives
+
+```text
+tr(P_s P_t) = (2/3)N       for s != t.
+```
+
+To see the normalization, the coefficient at the relative route indexed
+by p in B_s B_t* is r_p r_p^T. Distinctness makes the identity trace of
+B_s B_t* B_t B_s* equal sum_p ||r_p||^4 N = 1536N.
+Divide by 48^2 to obtain (2/3)N.
+
+The distinct-stage triple has the ordered trace
+
+```text
+tr(P_0 P_1 P_2) = tr(B_0 B_1* B_1 B_2* B_2 B_0*) / 48^3.
+```
+
+Its group word for ports p,q,r is
+
+```text
+g_1(p)g_1(q)g_2(q)g_2(r).
+```
+
+It is identity exactly when g_1(p)g_1(q)=g_2(r)g_2(q).
+Section 1 forces p=q=r. The scalar trace weight is
+(r_p^T r_q)(r_q^T r_r)(r_r^T r_p), hence
+
+```text
+tr(P_0 P_1 P_2) = 96 * 4^3 N / 48^3 = N/18.
+```
+
+Cyclic invariance of trace and transpose give the same value for all
+six distinct-stage orderings. In the cubic expansion, three words use
+one stage, eighteen use exactly two, and six use three. Therefore
+
+```text
+tr(T)   = 24N,
+tr(T^2) = 24N + 6(2/3)N = 28N,
+tr(T^3) = 24N + 18(2/3)N + 6(1/18)N = (109/3)N.
+```
+
+## 3. A spectral polynomial
+
+All eigenvalues of T lie in [0,3], because it is a sum of three
+orthogonal projections. Use the rational polynomial
+
+```text
+p(t) = 1 - (1 - 12t/13)^2 (1 - t/3)
+     = (85/39)t - (248/169)t^2 + (48/169)t^3.
+```
+
+It has p(0)=0 and p(t)<=1 throughout [0,3]. Summing over the nonzero
+eigenvalues gives rank(T)>=tr(p(T)). The exact moments imply
+
+```text
+rank([B0;B1;B2]) = rank(T)
+ >= [(85/39)24 - (248/169)28 + (48/169)(109/3)] N
+  = (280/13)N.
+```
+
+This improves the two-moment bound (144/7)N. It is still a lower bound,
+not an exact rank calculation. Rational and real matrix ranks agree.
+
+## 4. Transfer to the selected-input defect
+
+This is PR2153's defect argument, restated to identify the inherited
+contract. Write P_(s,A), P_(s,T) and ell_(s,A), ell_(s,T) for the
+ordered selected sections and their two-sided inverses. For h in ker B1
+put only pair 0 nonzero and set
+
+```text
+X_U=h,       Y_U=0,
+X_A=-P_(0,A)B0h-P_(2,A)B2h,          Y_A=-X_A,
+X_T=P_(0,T)ell_(0,A)P_(2,A)B2h,      Y_T=-X_T.
+```
+
+The original selected defect J(h)=(X_A,X_T) determines both queries:
+
+```text
+B0h=-ell_(0,A)X_A-ell_(0,T)X_T,
+B2h=ell_(2,A)P_(0,A)ell_(0,T)X_T.
+```
+
+Conversely, the two queries determine J(h). Since rank(B1)=8N,
+
+```text
+dim J(ker B1)=rank([B0;B1;B2])-8N >= ceil(176N/13).
+```
+
+The ordered F0 query is zero. After F0 the ordinary fields are X_U=Y_U=h,
+the selected A fields are opposite P_(2,A)B2h, and both T fields are zero.
+At B_1_2 the future query is zero, the logger clears its A output,
+and the current query is B1h=0. Only Y_(0,U)=h remains.
+
+The retained witness proof and independent rational review establish
+that the ordinary tail moves h into the former-Y hole. All selected
+values after B1, completed-X descendants and five forward histories
+are zero. Later homogeneous selected-only controls preserve zero.
+The formal-h tail is inherited evidence, not rerun by this package.
+There is no separately retained transient F0 A output or free read
+of the former-Y hole in this observation contract.
+
+Restrict an additional rational-linear record to this family, obtaining
+C of rank at most d. Linear algebra gives
+
+```text
+dim J(ker C) >= dim J(ker B1) - rank(C).
+```
+
+A complement of ker C has dimension rank(C), so its image can account
+for at most that many defect dimensions. A positive residual supplies
+an input with zero observations but nonzero required selected output.
+No decoder, even nonlinear, can distinguish it from zero. Fixed offsets
+and common arbitrary dirty baselines do not change this comparison.
+
+For q scalars per owner, capacity is at most qN on this subspace.
+At q=13 the residual is at least ceil(7N/13)>0. An eight-coordinate
+record, including the full original R2 X query, leaves at least
+ceil(72N/13) invisible dimensions. Extra independent payload storage
+must count in d. These are information dimensions, not paid ranks.
+
+## 5. Limits
+
+The finite route facts and exact arithmetic are checked by the new
+verifier. The spectral and defect implications are written proofs,
+not proof-assistant formalizations. The global erased-family trajectory
+and local packet-to-native binding remain inherited from PR1969.
+
+Records of width fourteen or fifteen remain unresolved. PR2174's
+sixteen-coordinate logical recovery also grants selected values at
+the B1 cut and does not supply a final-only physical controller.
+Changed first pairs or schedules, nonlinear record encoders and a
+complete native invoice lie outside this result. No supplier admission,
+recursive precision bound or multiplication exponent change follows.

research/qq16-third-moment-capacity/README.md

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+# QQ16 third-moment transcript capacity
+
+Thirteen rational query coordinates per regular-cover owner cannot repair
+the fixed QQ16 selected-input loss under the retained erased-family
+observation contract. The new certificate gives
+
+```text
+rank([B0;B1;B2]) >= 280|G|/13,
+selected A/T defect >= ceil(176|G|/13),
+residual after a width-13 record >= ceil(7|G|/13).
+```
+
+Fourteen coordinates are necessary, not proved sufficient or optimal.
+These are information dimensions, not paid recursive ranks. The selected
+frontier and all frozen packages remain unchanged. No new supplier or κ.
+
+## What changed
+
+Ross Richardson's [PR2104](https://github.com/CrocSwap/integer-mult-bounds/pull/2104)
+uses two exact moments to obtain rank at least 144|G|/7.
+commdata2338's [PR2153](https://github.com/CrocSwap/integer-mult-bounds/pull/2153)
+transfers that bound to the joint A/T defect and excludes records with at
+most twelve coordinates per owner.
+
+This package adds the third moment, `tr(T^3)=(109/3)|G|`, for
+`T=sum_s B_s*B_s/48`, and an exact cubic spectral certificate. It improves
+their defect lower bound by 88|G|/91 and excludes width thirteen.
+[PROOF.md](PROOF.md) includes the ordered route and defect arguments.
+
+Two different checks establish the new finite premise. A complete census
+of 18,432 ordered route products finds that the stage-1 and stage-2
+product sets intersect only at identity. A chart-swap reconstruction
+checks the same implication using disjoint external axes and 96 distinct
+ordinary frames. Their identity triple coefficient is exactly 6,144.
+No predecessor executable is imported.
+
+## Reproduce
+
+Python 3.12, standard library only, from the repository root:
+
+```sh
+python3.12 -I -B scripts/research_package.py --package research/qq16-third-moment-capacity
+python3.12 -I -O -B research/qq16-third-moment-capacity/verify.py
+python3.12 -I -O -B research/qq16-third-moment-capacity/test_verify.py
+```
+
+Expected verifier line:
+
+```text
+PASS_QQ16_THIRD_MOMENT_CAPACITY rank>=280|G|/13 defect>=176|G|/13 width>=14 no_new_kappa
+```
+
+Normal and optimized registered replay pass on Linux with CPython 3.12.15.
+The direct normal/optimized verifier runs also pass under a 512 MiB
+address-space cap, with a measured peak child RSS of 114,672 KiB. The Git
+provenance launcher ran without that cap because Git maps a larger existing
+packfile. All 65 research-workflow tests and selected-record integrity pass.
+
+The read-only verifier checks live source hashes and exact equality with
+[RESULT.json](RESULT.json). Three semantic controls reject a changed query
+coefficient, collapsed stage routes and a duplicate chart. Seven behavior
+and arithmetic tests include source and receipt corruption through the CLI.
+Validation stays enabled under `-O`.
+
+Source commits, Git blobs and hashes are in [SOURCES.json](SOURCES.json).
+[TASK.json](TASK.json) records the central-handoff override, overlap,
+predecessor heads, scope and final freshness check. The generic launcher
+also authenticates historical task and selected-record provenance.
+
+## Verification limits
+
+This is author-checked exact finite source evidence plus a written proof.
+It has no independent human or formal-kernel review. The retained QQ16
+witness supplies the global erased-family trajectory; this package does
+not rerun it, the native local words or the full selected cold replay.
+
+Records of width fourteen and fifteen, nonlinear encoders and changed
+trajectories remain outside the exclusion. Native access, retention,
+cleanup, ownership, tape, recursive precision and supplier adoption remain
+open. PR2174's adjacent sixteen-coordinate recovery grants the B1-cut
+selected values; it is not a final-only controller or paid-cost invoice.
+
+## Implementation bounds
+
+The verifier has no recursion or dynamic code evaluation. Its route
+products have exactly 96² iterations per stage; binary application has
+at most 25 iterations. The 48 spectator axes must remain fixed. Packet
+input is at most 1 MiB compressed and 16 MiB decompressed, and every
+proof-relevant collection comes from fixed, hash-bound source bytes.
+Functions stay below 60 physical lines and all semantic guards remain
+active under optimization. Each test subprocess has a 30-second timeout.
+
+Power-of-ten rule 3 has a managed-runtime deviation in `verify.py` and
+`test_verify.py`: Python allocates parsed objects, integers and census
+tuples during checking. The fixed source dimensions, input caps and
+18,432-product limit bound this batch workload. Replay under a memory cap
+checks the compensating control. A fixed-capacity native implementation
+would only be needed if this certificate becomes a larger search tool.
+The diagnostics are `-Wall` compilation and normal/optimized tests. No
+additional Python static analyzer was run.
+
+## Open next questions
+
+1. Can a higher-moment certificate exclude width fourteen or fifteen?
+   The exact stack rank determines the remaining information gap. First
+   compute another identity trace with all route products counted, then
+   bound the spectral indicator on [0,3]. The cover is too large for naive
+   matrix expansion, and additional moments may still not decide the gap.
+2. Can a proposed width-fourteen or width-fifteen record recover the
+   selected inputs under the complete observation operator? Test its
+   kernel against all required selected rows before native costing.
+   No such record or final-only controller has been supplied. The
+   sixteen-coordinate result in PR2174 is a useful upper description
+   under its explicitly granted B1-cut observations.
+
+## Credit
+
+Nick Petalas. New work is Apache-2.0; retained notices are unchanged.
+Ali Romman and the PR1969 contributors supplied the conserved maps,
+selected sections, route packet and erased witness. Ross Richardson
+supplied the two-moment method in PR2104. commdata2338 supplied the joint
+A/T capacity argument in PR2153 and the adjacent recovery in PR2174.
+Cinna Davis's PR2079/PR2122 and Adam Gall's PR2086/PR2093 supplied the
+preceding query and logger investigations. The inherited supplier lineage
+includes Rohan Arun, Jeremy Kothe, Hao Yan and Zhihao Chen.

research/qq16-third-moment-capacity/RESULT.json

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+{
+  "route_census": {
+    "ordered_pairs_per_stage": 9216,
+    "distinct_products_per_stage": [
+      9121,
+      9121
+    ],
+    "intersection_size": 1,
+    "identity_pairs_per_stage": 96
+  },
+  "independent_distinct_swap_charts": 96,
+  "cross_trace_coefficient": 1536,
+  "triple_trace_coefficient": 6144,
+  "projector_moments_per_owner": [
+    "24",
+    "28",
+    "109/3"
+  ],
+  "spectral_polynomial_coefficients": [
+    "0",
+    "85/39",
+    "-248/169",
+    "48/169"
+  ],
+  "stacked_rank_lower_per_owner": "280/13",
+  "selected_defect_lower_per_owner": "176/13",
+  "width_thirteen_residual_lower_per_owner": "7/13",
+  "necessary_integer_record_width": 14,
+  "semantic_controls_rejected": 3,
+  "status": "PASS_QQ16_THIRD_MOMENT_CAPACITY",
+  "scope": "Regular-cover information dimension; inherited erased-family contract; no native invoice or new kappa."
+}