Rule out fifteen-coordinate QQ16 transcripts via the fourth moment

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

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+# QQ16: fifteen-coordinate linear records still lose selected inputs
+
+For the fixed QQ16 `F0/B_1_2` erasure and observation contract, every
+read-only rational-linear record with at most **fifteen scalars per
+regular-cover owner fails** to restore the original selected inputs.
+The new exact fourth moment gives
+
+```text
+tr(T^4) = (1823/36)|G|,
+rank([B0;B1;B2]) >= (117/5)|G|,
+selected A/T defect dimension >= ceil((77/5)|G|).
+```
+
+A rank-d record leaves at least `max(0,ceil(77|G|/5)-d)` selected
+input dimensions invisible. At width fifteen the residual is at least
+`ceil(2|G|/5)>0`. Thus an integer number of coordinates per owner must
+be at least **sixteen**, strengthening the necessary width fourteen
+from Nick Petalas's [PR #2181](https://github.com/CrocSwap/integer-mult-bounds/pull/2181).
+This is a scoped proof with exact finite certificates, author-checked
+by two different calculations. It is not an independently reviewed or
+formalized theorem, a physical cost bound, or a new supplier or kappa.
+
+The claim is in [discussion #2074](https://github.com/CrocSwap/integer-mult-bounds/discussions/2074#discussioncomment-18935517),
+following #2181's higher-moment handoff. The record encoder is linear;
+the decoder may be arbitrary. Fixed offsets and common dirty baselines
+are allowed. Changed first pairs, changed ordinary trajectories,
+nonlinear encoders and uncounted independent payload storage are excluded.
+
+## Reproduce
+
+Python 3.12, standard library only, from the repository root:
+
+```sh
+python3 -I -B scripts/research_package.py --package research/qq16-fourth-moment-capacity
+```
+
+The hook runs the primary certificate, the separate block-Gram oracle,
+and the semantic/artifact controls in normal and optimized modes. Direct
+read-only replay, also usable in a source export:
+
+```sh
+python3 -I -O -B research/qq16-fourth-moment-capacity/verify.py
+python3 -I -O -B research/qq16-fourth-moment-capacity/oracle.py
+```
+
+Expected lines include:
+
+```text
+PASS_QQ16_FOURTH_MOMENT_CAPACITY moment4=1823/36 rank>=117|G|/5 defect>=77|G|/5 width>=16 no_new_kappa
+PASS_QQ16_BLOCK_GRAM_ORACLE moments=24,28,109/3,1823/36 blocks=9
+```
+
+[RESULT.json](RESULT.json) contains the exact pair-product census, its
+six full-matrix digests, all stage-word counts and rational coefficients.
+[SOURCES.json](SOURCES.json) pins the immutable main inputs; pending PR
+checkouts, network access and third-party modules are unnecessary.
+[TASK.json](TASK.json) records the scope change and exact predecessor heads.
+
+## 1. Operators and finite premises
+
+Use the actual 96 ordinary ports U and the three query operators
+`B_s=R_s iota_U`, for s=0,1,2, of Ali Romman's retained
+[joint-query package](../joint-query/README.md). Let G be a finite group
+containing the literal owner-route matrices, acting regularly, and put
+N=|G|. As in the inherited witness, restrict to fields constant in each
+payload block: payload translations disappear, but arbitrary owner
+values remain. This is not a quotient that identifies distinct routes.
+The packet-to-native interpretation remains inherited.
+
+For ordinary port p, write r_p for its eight-component integer column
+and g_s(p) for its owner route. The pinned packet checks establish
+
+```text
+g_0(p)=1,     g_s(p)^2=1,
+r_p^T r_p=4,                 sum_p r_p r_p^T=48 I_8.
+```
+
+The stage-1 and stage-2 routes act on the first 25 binary axes and fix
+the remaining 48 of the original 73 axes. The shared source binder
+checks the entire 73-column input, the common orthonormal chart-swap
+form, the A/T/U partition, both selected Grams, and agreement between
+F1/F2 and the current/future B_1_2 alignment maps. Its extraction and
+chart checks are adapted, with credit, from #2181.
+
+Let `h_st(p)=g_s(p)g_t(p)^-1` and
+`C_st(p,q)=h_st(p)h_st(q)^-1`, for each ordered s!=t. A new exhaustive
+exact binary-matrix census checks all **6*96^2 = 55,296** ordered products:
+
+* Each of the six tables has 9,121 distinct products. Its identity
+  fibre is precisely the 96 diagonal pairs `(p,p)`; every other fibre
+  is a singleton.
+* For every s, with t and u the other two stages, the images of C_st
+  and C_su intersect only in the identity.
+
+All port pairs are included, even when their scalar coefficient is zero.
+Matrix equality is on all active axes with the spectators checked as
+fixed. No enumeration of the huge group G or probabilistic hashing is
+used for equality; the digests are portable receipts of exact tables.
+
+Under the regular right action `Q_g f(d)=f(dg)`, `tr Q_g` is N for the
+identity and zero otherwise. Put
+
+```text
+P_s=B_s*B_s/48,        T=P_0+P_1+P_2.
+```
+
+Since `B_s B_s*=48 I_(8N)`, each P_s is an orthogonal projection of
+rank 8N. Moreover `rank T=rank [B0;B1;B2]`. All matrices are rational,
+so real and rational ranks agree.
+
+## 2. The fourth trace
+
+Set `M_st=B_s B_t*` and `A_st=M_st M_ts`. The coefficient of A_st at
+route C_st(p,q) is contributed by
+
+```text
+(r_p^T r_q) r_p r_q^T.
+```
+
+At the identity, summing the diagonal fibre gives `192 I_8`. All other
+coefficients have a unique ordered pair. For a self-adjoint group-algebra
+matrix A, its regular trace satisfies
+
+```text
+tr(A^2)/N = sum_g ||A[g]||_F^2.
+```
+
+Here the full root Gram square sum is
+`sum_(p,q) (r_p^T r_q)^2 = tr((48 I_8)^2)=18432`; the diagonal contributes
+`96*16=1536`. Therefore, for s!=t,
+
+```text
+tr(P_s P_t P_s P_t)/N
+ = [8*192^2 + 16*(18432-1536)]/48^4
+ = 565248/5308416 = 23/216.                         (1)
+```
+
+For distinct s,t,u the two A tables have only the identity in common.
+The group-coefficient Frobenius inner product consequently gives
+
+```text
+tr(P_s P_t P_s P_u)/N = 8*192^2/48^4 = 1/18.       (2)
+```
+
+The lower moments retain #2104/#2181's argument. Relative routes are
+distinct, so `tr(P_s P_t)/N=1536/48^2=2/3`. In a distinct-stage triple,
+the identity condition is
+`g_1(p)g_1(q)=g_2(r)g_2(q)`. The checked product intersection and identity
+fibres force p=q=r. Thus the triple trace is `96*4^3/48^3=1/18`.
+Cyclic invariance and transpose cover its six orderings.
+
+Cyclic removal of adjacent repeated projections partitions all 81
+stage words of length four as follows. These are traces, so the first
+and last letters also count as adjacent.
+
+| Reduced type | Count | Trace divided by N |
+|---|---:|---:|
+| one stage | 3 | 8 |
+| two adjacent stages | 36 | 2/3 |
+| three distinct stages | 24 | 1/18 |
+| alternating `s,t,s,t` | 6 | 23/216 |
+| repeated-stage `s,t,s,u`, all three stages used | 12 | 1/18 |
+
+Their sum is `1823/36`. The complete first four moments are
+
+```text
+(tr T, tr T^2, tr T^3, tr T^4)/N = (24,28,109/3,1823/36).   (3)
+```
+
+### Separate contraction oracle
+
+[oracle.py](oracle.py) does not assume the collision classification,
+use the primary binary multiplication, or reduce cyclic stage words.
+It shares only the credited source binder. It converts routes to binary
+row matrices and explicitly forms every group coefficient of the nine
+blocks of `C^2`, where `C=(B_s B_t*)_(s,t)` is the 24N-dimensional
+block Gram matrix. The two rank-one factors multiply as
+`r_p(r_p^T r_q)r_q^T`; equal-stage blocks are `48I`.
+
+Because C is self-adjoint, coefficient Frobenius contractions compute
+`tr(C^2)`, `tr(C^3)=<C,C^2>` and `tr(C^4)=||C^2||_F^2`. Dividing by
+48^k gives (3), since C/48 and T have the same nonzero eigenvalues.
+For the fourth trace each diagonal block contributes 1187/108 and each
+off-diagonal block 53/18. Their total is again 1823/36. This is a second
+implementation and method, not an independent human review.
+
+## 3. Rational spectral certificate
+
+For every real t,
+
+```text
+p(t) = 1-(1-81t/40+9t^2/10)^2 <= 1,       p(0)=0
+     = (81/20)t-(9441/1600)t^2+(729/200)t^3-(81/100)t^4.
+```
+
+Apply this inequality to each nonzero eigenvalue of T. The zero
+eigenvalues contribute zero. Equation (3) gives
+
+```text
+rank T >= tr p(T) = (117/5)N.                         (4)
+```
+
+No floating-point extremization is used for acceptance. The coefficients
+come from the exact 2x2 moment system
+`[[28,109/3],[109/3,1823/36]] [a,b]^T=[24,28]^T`, giving
+`a=81/40,b=-9/10`, then `p=1-(1-a t-b t^2)^2`. The checker verifies the
+polynomial identity and the rational trace evaluation exactly.
+
+## 4. Inherited selected-input implication
+
+This is the joint-defect argument of this account's
+[PR #2153](https://github.com/CrocSwap/integer-mult-bounds/pull/2153),
+restated to make the contract explicit. For h in `V=ker B1`, use only
+pair 0 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.
+```
+
+All other banks and helpers are zero. The original selected defect
+`J(h)=(X_A,X_T)` determines and is determined by `(B0h,B2h)`:
+
+```text
+B0h=-ell_(0,A)X_A-ell_(0,T)X_T,
+B2h=ell_(2,A)P_(0,A)ell_(0,T)X_T.
+```
+
+The two-sided selected inverse relations preserve the displayed order.
+Consequently
+
+```text
+dim J(V)=rank([B0;B1;B2])-rank B1 >= ceil(77N/5).       (5)
+```
+
+The original QQ16 witness establishes, for every such h, zero F0 query
+and zero selected state after B1, with only Y_(0,U)=h remaining. Its
+retained ordinary tail moves h into the former-Y hole. The allowed old
+observations—selected values after B1 and later selected-only descendants,
+completed-X descendants, and all five forward histories—are zero.
+An independently saved transient F0 A value or a read of the former-Y
+hole is additional information, not part of those old observations.
+This trajectory and native-frame interpretation are inherited from the
+pinned witness and rational review, not newly replayed by this package.
+
+If C is the restriction of an added record to V and rank C<=d, then
+`dim J(ker C)>=dim J(V)-d`: a complement of ker C has dimension at most d
+and its J-image adds at most d dimensions. Combining with (5) proves
+the stated residual. Positive residual gives two inputs with identical
+records and different required selected outputs; no decoder can
+separate them. Common dirty baselines and fixed affine offsets cancel
+when comparing the two executions.
+
+A q-coordinate record with one rational scalar per owner has capacity
+at most qN. Every integer q<=15 fails. The inherited sixteen-coordinate
+record `(B0 X_U,B2 X_U)` recovers this particular erased family, so its
+integer-coordinate threshold is sixteen. This does not compute its
+exact defect dimension, which lies between `ceil(77N/5)` and 16N.
+
+The lower bound also applies when all post-B1 selected values are
+granted, as in [PR #2174](https://github.com/CrocSwap/integer-mult-bounds/pull/2174).
+It therefore matches that PR's sixteen-coordinate logical upper bound
+for pair 0 **under its explicit recorded-cut premise**. Its full-input
+recovery proof is credited, not rerun or promoted to a final-only
+controller here. Sixteen information coordinates are not sixteen paid
+recursive ranks.
+
+## Verification limits
+
+The finite route facts, all four moments, polynomial identity and
+rejection controls are executed. The spectral and capacity arguments
+are written proofs. The two implementations share the source binder;
+no formal kernel or independent human review is claimed. No full QQ16
+native replay, selected frontier cold replay, final-only controller,
+compact actual-cover null vector, precision bound, chronological hosting,
+physical record implementation or completed supplier bill was checked.
+All source packages and the selected record remain unchanged.
+
+## Open next questions
+
+1. **Can an actual final-only controller recover the post-B1 selected
+   cut values used by #2174?** Trace one literal selected program and
+   express its cut values in its final observations. This is the missing
+   access premise; the integer width gap for the erased family no longer
+   motivates more moment searches.
+2. **What is the completed native bill for the sixteen-coordinate
+   record and #2174's decoder?** Start by jointly pricing both pre-F0
+   ordinary reads and the completed-X B2 read in their actual endpoint
+   frames. Record retention, dirty restoration, hosting, tape and
+   precision remain blockers. Stop construction if even the favorable
+   complete bill misses its target.
+
+## Credit
+
+Prepared for commdata2338 (@commdata2338), with substantial OpenAI Codex assistance.
+New material is Apache-2.0.
+
+Ali Romman and the contributors in [joint-query/NOTICE.md](../joint-query/NOTICE.md)
+supplied PR #1969's maps, ordered sections, actual routes and witness
+(head `a6e6ad3936f67b8f63b5b68c5e75d540610a65e7`). Ross Richardson's
+PR #2104 (`4bdc574da394a39f955e8cf4637075649c43a632`) supplied the
+rank-moment method. Nick Petalas's PR #2181
+(`e18ca76987f5945c6a4b5b6bebbe73cdab3c4561`) supplied the third moment,
+the polynomial approach and the source binder adapted here. This
+account's PR #2153 (`5646398dcdc01769df83b939c6b3eb5ecb4d5ebd`) supplied
+the joint A/T defect transfer; PR #2174
+(`f79470a7931f3943360c10d15b428e0e4e479c22`) supplies the distinct
+recorded-cut recovery upper bound. Cinna Davis's #2079/#2122 and Adam
+Gall's #2086/#2093 are adjacent observation obstructions. Rohan Arun,
+Jeremy Kothe and the retained geometry/native/hosting lineage keep
+their existing attribution. The new contribution is the fourth moment,
+its separate contraction oracle, and the stronger transcript bound.